2923:
right (left) modules over a ring, is one example. The fact that the
Jacobson radical can be viewed as the intersection of all maximal right (left) ideals in the ring, shows how the internal structure of the ring is reflected by its modules. It is also a fact that the intersection of all maximal right
1088:
are quite different in flavour, since more unusual behavior can arise. While the theory has developed in its own right, a fairly recent trend has sought to parallel the commutative development by building the theory of certain classes of noncommutative rings in a geometric fashion as if they were
3172:. Central to the development of these subjects were the rings of integers in algebraic number fields and algebraic function fields, and the rings of polynomials in two or more variables. Noncommutative ring theory began with attempts to extend the complex numbers to various
1897:
1070:, a major area of modern mathematics. Because these three fields (algebraic geometry, algebraic number theory and commutative algebra) are so intimately connected it is usually difficult and meaningless to decide which field a particular result belongs to. For example,
2718:
2031:
2547:
1657:
2181:
2892:
rings that contain no non-trivial proper (two-sided) ideals, yet contain non-trivial proper left or right ideals. Various invariants exist for commutative rings, whereas invariants of noncommutative rings are difficult to find. As an example, the
2907:
matrices over a division ring never forms an ideal, irrespective of the division ring chosen. There are, however, analogues of the nilradical defined for noncommutative rings, that coincide with the nilradical when commutativity is assumed.
2368:
of commutative rings. However, commutative rings can be Morita equivalent to noncommutative rings, so Morita equivalence is coarser than isomorphism. Morita equivalence is especially important in algebraic topology and functional analysis.
1952:
1799:
1235:. This correspondence has been enlarged and systematized for translating (and proving) most geometrical properties of algebraic varieties into algebraic properties of associated commutative rings.
3176:
systems. The genesis of the theories of commutative and noncommutative rings dates back to the early 19th century, while their maturity was achieved only in the third decade of the 20th century.
2214:
2617:
592:
2469:
1957:
884:
2753:
1591:
1166:, non-trivial commutative rings where no two non-zero elements multiply to give zero, generalize another property of the integers and serve as the proper realm to study divisibility.
2120:
2084:
2585:
2456:
1756:
837:
800:
2842:
2410:
2242:
2112:
2055:
1791:
546:
3158:
2924:
ideals in a ring is the same as the intersection of all maximal left ideals in the ring, in the context of all rings; irrespective of whether the ring is commutative.
2897:, the set of all nilpotent elements, is not necessarily an ideal unless the ring is commutative. Specifically, the set of all nilpotent elements in the ring of all
365:
1259:), and a general scheme is then obtained by "gluing together" (by purely algebraic methods) several such affine schemes, in analogy to the way of constructing a
3131:
3056:
1712:
989:
3075:
2033:
that is maximal in the sense that it is impossible to insert an additional prime ideal between two ideals in the chain, and all such maximal chains between
1064:, which provide many natural examples of commutative rings, have driven much of the development of commutative ring theory, which is now, under the name of
2086:
have the same length. Practically all noetherian rings that appear in applications are catenary. Ratliff proved that a noetherian local integral domain
1389:
over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by
1333:(integral domains in which every non-zero element is invertible) act on vector spaces. Examples of noncommutative rings are given by rings of square
1142:. Commutative rings resemble familiar number systems, and various definitions for commutative rings are designed to formalize properties of the
3858:
446:
1425:, in which elements of a group are represented by invertible matrices in such a way that the group operation is matrix multiplication.
2322:
is a regular local ring if and only if it has finite global dimension and in that case the global dimension is the Krull dimension of
1514:
1082:, which is a part of commutative algebra, but its proof involves deep results of both algebraic number theory and algebraic geometry.
358:
3013:. Those rings are essentially the same things as varieties: they correspond in essentially a unique way. This may be seen via either
3840:
3821:
3795:
3749:
3715:
3689:
3659:
3621:
3557:
3523:
3505:
3475:
3440:
1284:
982:
934:
1042:), as well as an array of properties that proved to be of interest both within the theory itself and for its applications, such as
1892:{\displaystyle \textstyle \operatorname {gr} _{\mathfrak {m}}(R)=\bigoplus _{k\geq 0}{\mathfrak {m}}^{k}/{{\mathfrak {m}}^{k+1}}}
2364:. In fact, two commutative rings which are Morita equivalent must be isomorphic, so the notion does not add anything new to the
1074:
is a theorem which is fundamental for algebraic geometry, and is stated and proved in terms of commutative algebra. Similarly,
1918:
1523:
351:
2713:{\displaystyle 1\to R^{*}\to F^{*}{\overset {f\mapsto fR}{\to }}\operatorname {Cart} (R)\to \operatorname {Pic} (R)\to 1}
1495:
1453:
1170:
are integral domains in which every ideal can be generated by a single element, another property shared by the integers.
3710:, Cambridge Studies in Advanced Mathematics, vol. 8 (Second ed.), Cambridge, UK.: Cambridge University Press,
3014:
2189:
1220:
1071:
1039:
1023:
975:
28:
3291:
3067:
3010:
1191:
692:
220:
1722:. The fundamental theorem of dimension theory states that the following numbers coincide for a noetherian local ring
1542:
1485:
2927:
Noncommutative rings are an active area of research due to their ubiquity in mathematics. For instance, the ring of
2542:{\displaystyle \operatorname {Spec} R\to \mathbb {Z} ,\,{\mathfrak {p}}\mapsto \dim M\otimes _{R}k({\mathfrak {p}})}
2026:{\displaystyle {\mathfrak {p}}={\mathfrak {p}}_{0}\subsetneq \cdots \subsetneq {\mathfrak {p}}_{n}={\mathfrak {p}}'}
3549:
3497:
2803:
1567:
452:
556:
3278:
2916:
1652:{\displaystyle {\mathfrak {p}}_{0}\subsetneq {\mathfrak {p}}_{1}\subsetneq \cdots \subsetneq {\mathfrak {p}}_{n}}
1018:
in which addition and multiplication are defined and have similar properties to those operations defined for the
3204:
2990:
1463:
1047:
927:
851:
730:
680:
2726:
2311:
2176:{\displaystyle \operatorname {dim} R=\operatorname {ht} {\mathfrak {p}}+\operatorname {dim} R/{\mathfrak {p}}}
1243:, a generalization of algebraic varieties, which may be built from any commutative ring. More precisely, the
1075:
1302:
1288:
1175:
1116:
1094:
1061:
739:
432:
311:
3224:
3180:
1236:
1187:
1167:
1090:
896:
747:
698:
479:
2552:
2423:
1725:
3228:
3192:
1402:
1378:
1355:
1350:
2060:
813:
776:
3884:
3247:
3071:
2935:
2894:
2816:
2384:
2256:
2223:
2217:
2093:
2036:
1772:
1527:
1489:
1422:
1390:
1386:
1374:
1334:
1318:
1294:
1240:
1027:
620:
494:
298:
290:
262:
257:
248:
205:
147:
3518:, Mathematical Surveys and Monographs, vol. 65, Providence, RI: American Mathematical Society,
3266:
3173:
2885:
2879:
2772:
2327:
2287:
1531:
1472:
1442:
1437:
1414:
1410:
1394:
1366:
1330:
1322:
1306:
1280:
1252:
1151:
1129:
1093:
on (non-existent) 'noncommutative spaces'. This trend started in the 1980s with the development of
1085:
1066:
1043:
1015:
902:
710:
661:
606:
500:
486:
414:
382:
316:
306:
157:
57:
49:
40:
2950:
of abelian groups are rarely commutative, the simplest example being the endomorphism ring of the
529:
3853:, Graduate Studies in Mathematics, vol. 145, Providence, RI: American Mathematical Society,
3270:
3208:
3136:
3006:
2868:
2339:
2315:
2268:
1900:
1298:
1244:
1216:
1211:
1147:
1110:
1057:
1011:
915:
473:
401:
122:
113:
71:
3736:, Graduate Studies in Mathematics, vol. 30, Providence, RI: American Mathematical Society,
3767:
3854:
3836:
3817:
3791:
3745:
3711:
3685:
3655:
3617:
3553:
3519:
3501:
3471:
3436:
3239:
3220:
2947:
2413:
2353:
1224:
956:
753:
518:
459:
3467:
3461:
3737:
3647:
3609:
3584:
3282:
3169:
3026:
2975:
2951:
2912:
2810:
2795:
2756:
2291:
1363:
1268:
1264:
1256:
1248:
1199:
1183:
1171:
1053:
962:
948:
762:
704:
667:
467:
440:
426:
142:
3868:
3831:
3805:
3759:
3725:
3699:
3669:
3631:
3567:
3533:
3485:
1178:
can be carried out. Important examples of commutative rings can be constructed as rings of
3864:
3827:
3801:
3787:
3755:
3721:
3695:
3681:
3665:
3643:
3627:
3605:
3563:
3529:
3481:
3286:
2998:
2365:
1577:
1499:
1476:
1457:
1398:
1314:
1232:
1219:
is in many ways the mirror image of commutative algebra. This correspondence started with
1195:
1163:
1102:
1101:. It has led to a better understanding of noncommutative rings, especially noncommutative
724:
674:
512:
234:
228:
215:
195:
186:
152:
89:
842:
3780:
3542:
3168:
Commutative ring theory originated in algebraic number theory, algebraic geometry, and
3081:
3032:
1662:
1467:
768:
276:
3587:(1981), "Emmy Noether and Her Influence", in Brewer, James W; Smith, Martha K (eds.),
3878:
3592:
3432:
3312:
3251:
2920:
1912:
1518:
1508:
1503:
1228:
1098:
1035:
909:
805:
420:
162:
127:
84:
3058:
that are invariant under the action of a finite group (or more generally reductive)
3424:
3262:
3258:
3200:
3188:
2768:
2592:
1546:
1511:
consists of theorems determining when two rings have "equivalent" module categories
1382:
1326:
1310:
1159:
941:
716:
612:
336:
267:
101:
3848:
1337:
or more generally by rings of endomorphisms of abelian groups or modules, and by
3235:
3216:
2958:
2889:
1550:
1479:
1418:
1359:
1338:
1155:
1139:
921:
632:
506:
326:
321:
210:
200:
167:
3246:(1928). Wedderburn's structure theorems were formulated for finite-dimensional
3651:
3613:
3243:
3184:
2888:
is more complicated than that of a commutative ring. For example, there exist
1179:
1079:
1031:
686:
76:
3642:, Graduate Texts in Mathematics, vol. 131 (Second ed.), New York:
3196:
646:
551:
331:
137:
94:
62:
3816:, Pure and Applied Mathematics, vol. 127, Boston, MA: Academic Press,
1108:
For the definitions of a ring and basic concepts and their properties, see
17:
3516:
Rings and Things and a Fine Array of
Twentieth Century Associative Algebra
3285:
called this work "revolutionary"; the publication gave rise to the term "
2939:
1260:
640:
626:
132:
2806:) that measures the deviation of the ring of integers from being a PID.
2771:
domain (i.e., regular at any prime ideal), then Pic(R) is precisely the
2943:
1406:
1143:
1114:. The definitions of terms used throughout ring theory may be found in
1019:
1003:
524:
408:
3741:
3548:, London Mathematical Society Student Texts, vol. 16, Cambridge:
3074:
are polynomials that are invariant under permutation of variable. The
2318:
is an example of a Cohen–Macaulay ring. It is a theorem of Serre that
1223:
that establishes a one-to-one correspondence between the points of an
3025:
A basic (and perhaps the most fundamental) question in the classical
1247:
of a commutative ring is the space of its prime ideals equipped with
66:
1255:
of rings. These objects are the "affine schemes" (generalization of
3261:, in collaboration with W. Schmeidler, published a paper about the
1421:. The most prominent of these (and historically the first) is the
27:
This article is about a mathematical concept. For other uses, see
2373:
Finitely generated projective module over a ring and Picard group
3680:, Problem Books in Mathematics (Second ed.), New York:
1150:. In commutative ring theory, numbers are often replaced by
3575:
3273:. The following year she published a landmark paper called
2326:. The significance of this is that a global dimension is a
3604:, Graduate Texts in Mathematics, vol. 189, New York:
3494:
Groups, Rings and Fields: Algebra through practice, Book 3
3407:
3405:
2957:
One of the best-known strictly noncommutative ring is the
2798:, which is Dedekind and thus regular. It follows that Pic(
3786:, Graduate Texts in Mathematics, vol. 88, New York:
3215:. These noncommutative algebras, and the non-associative
3017:
or scheme-theoretic constructions (i.e., Spec and Proj).
2915:
of a ring; that is, the intersection of all right (left)
3029:
is to find and study polynomials in the polynomial ring
1362:
that draws heavily on non-commutative rings. It studies
1305:
based on noncommutative rings. Noncommutative rings and
3289:", and several other mathematical objects being called
3281:
with regard to (mathematical) ideals. Noted algebraist
1947:{\displaystyle {\mathfrak {p}}\subset {\mathfrak {p}}'}
3234:
The various hypercomplex numbers were identified with
1803:
3139:
3084:
3035:
2819:
2729:
2620:
2555:
2472:
2426:
2412:
the set of isomorphism classes of finitely generated
2387:
2226:
2192:
2123:
2096:
2063:
2039:
1960:
1921:
1802:
1775:
1728:
1665:
1594:
1056:
are much better understood than noncommutative ones.
854:
816:
779:
559:
532:
3766:
O'Connor, J. J.; Robertson, E. F. (September 2004),
2938:
is noncommutative despite its natural occurrence in
1022:. Ring theory studies the structure of rings, their
2360:is equivalent to the category of left modules over
2251:is an integral domain that is a finitely generated
3779:
3544:An Introduction to Noncommutative Noetherian Rings
3541:
3466:(Second ed.), Edward Arnold, London, p.
3227:types. One sign of re-organization was the use of
3152:
3125:
3050:
2836:
2747:
2712:
2603:is an integral domain with the field of fractions
2579:
2541:
2450:
2404:
2236:
2209:{\displaystyle \operatorname {ht} {\mathfrak {p}}}
2208:
2175:
2106:
2078:
2049:
2025:
1946:
1891:
1785:
1750:
1706:
1651:
878:
831:
794:
586:
540:
3850:The K-book: An introduction to algebraic K-theory
2090:is catenary if and only if for every prime ideal
3324:
1498:gives necessary and sufficient conditions for a
3223:before the subject was divided into particular
2946:and many parts of mathematics. More generally,
2458:subsets consisting of those with constant rank
1409:objects amenable to such a description include
1301:, attempts have been made recently at defining
3375:, Definition preceding Proposition 3.2 in Ch I
2298:is a noetherian local ring, then the depth of
1954:, there exists a finite chain of prime ideals
3835:. Vol. II, Pure and Applied Mathematics 128,
3540:Goodearl, K. R.; Warfield, R. B. Jr. (1989),
2611:, then there is an exact sequence of groups:
983:
359:
8:
3589:Emmy Noether: A Tribute to Her Life and Work
3076:fundamental theorem of symmetric polynomials
1769:The minimum number of the generators of the
587:{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }
2993:, then the set of all regular functions on
2981:The coordinate ring of an algebraic variety
3411:
3311:Ring theory may include also the study of
2302:is less than or equal to the dimension of
1488:determines the structure of a commutative
1146:. Commutative rings are also important in
990:
976:
377:
366:
352:
36:
3577:Abstract Algebra: Theory and Applications
3348:
3336:
3144:
3138:
3114:
3095:
3083:
3034:
2820:
2818:
2728:
2650:
2644:
2631:
2619:
2562:
2557:
2554:
2530:
2529:
2517:
2495:
2494:
2493:
2486:
2485:
2471:
2433:
2428:
2425:
2388:
2386:
2228:
2227:
2225:
2200:
2199:
2191:
2167:
2166:
2161:
2143:
2142:
2122:
2098:
2097:
2095:
2066:
2065:
2062:
2041:
2040:
2038:
2013:
2012:
2002:
1996:
1995:
1979:
1973:
1972:
1962:
1961:
1959:
1934:
1933:
1923:
1922:
1920:
1875:
1869:
1868:
1866:
1861:
1855:
1849:
1848:
1835:
1809:
1808:
1801:
1777:
1776:
1774:
1739:
1738:
1727:
1695:
1676:
1664:
1643:
1637:
1636:
1620:
1614:
1613:
1603:
1597:
1596:
1593:
1297:in many respects. Following the model of
879:{\displaystyle \mathbb {Z} (p^{\infty })}
867:
856:
855:
853:
823:
819:
818:
815:
786:
782:
781:
778:
580:
579:
571:
567:
566:
558:
534:
533:
531:
3732:McConnell, J. C.; Robson, J. C. (2001),
3009:, there is an analogous ring called the
2790:) vanishes. In algebraic number theory,
2748:{\displaystyle \operatorname {Cart} (R)}
1899:(equivalently, 1 plus the degree of its
1659:. It turns out that the polynomial ring
3772:MacTutor History of Mathematics Archive
3304:
1293:Noncommutative rings resemble rings of
380:
39:
3640:A First Course in Noncommutative Rings
3396:
3384:
3372:
3360:
3160:are elementary symmetric polynomials.
2970:The ring of integers of a number field
2844:; this results in a commutative ring K
2786:is a principal ideal domain, then Pic(
2286:Closely related concepts are those of
3492:Blyth, T.S.; Robertson, E.F. (1985),
2591:). It is an abelian group called the
2356:if the category of left modules over
7:
2255:-algebra, then its dimension is the
447:Free product of associative algebras
2580:{\displaystyle \mathbf {P} _{1}(R)}
2531:
2496:
2451:{\displaystyle \mathbf {P} _{n}(R)}
2229:
2201:
2168:
2144:
2099:
2067:
2042:
2014:
1997:
1974:
1963:
1935:
1924:
1870:
1850:
1810:
1778:
1751:{\displaystyle (R,{\mathfrak {m}})}
1740:
1638:
1615:
1598:
1325:that the ring acts on as a ring of
3678:Exercises in Classical Ring Theory
1915:if for every pair of prime ideals
1588:of all the chains of prime ideals
1557:Structures and invariants of rings
1517:gives insight on the structure of
1182:and their factor rings. Summary:
1174:are integral domains in which the
868:
25:
3231:to describe algebraic structure.
2874:Structure of noncommutative rings
1796:The dimension of the graded ring
1576:denotes a commutative ring. The
1285:Noncommutative algebraic geometry
1078:is stated in terms of elementary
935:Noncommutative algebraic geometry
3768:"The development of ring theory"
3431:, translated by Blocher, H. I.,
3250:while Artin generalized them to
2821:
2558:
2429:
2389:
2079:{\displaystyle {\mathfrak {p}}'}
1405:, which is non-commutative. The
1158:tries to capture the essence of
832:{\displaystyle \mathbb {Q} _{p}}
795:{\displaystyle \mathbb {Z} _{p}}
3734:Noncommutative Noetherian Rings
2837:{\displaystyle \mathbf {P} (R)}
2405:{\displaystyle \mathbf {P} (R)}
2259:of its field of fractions over
2237:{\displaystyle {\mathfrak {p}}}
2107:{\displaystyle {\mathfrak {p}}}
2050:{\displaystyle {\mathfrak {p}}}
1786:{\displaystyle {\mathfrak {m}}}
1584:is the supremum of the lengths
1562:Dimension of a commutative ring
1423:representation theory of groups
3325:Goodearl & Warfield (1989)
3120:
3088:
3045:
3039:
2831:
2825:
2742:
2736:
2704:
2701:
2695:
2686:
2683:
2677:
2659:
2652:
2637:
2624:
2574:
2568:
2536:
2526:
2501:
2482:
2445:
2439:
2399:
2393:
1825:
1819:
1745:
1729:
1701:
1669:
1438:Isomorphism theorems for rings
1313:) are often studied via their
1239:completed this by introducing
873:
860:
1:
3602:Lectures on Modules and Rings
3460:Allenby, R. B. J. T. (1991),
3275:Idealtheorie in Ringbereichen
3068:ring of symmetric polynomials
2856:(S) if two commutative rings
1040:universal enveloping algebras
1026:, or, in different language,
3706:Matsumura, Hideyuki (1989),
1475:determines the structure of
1466:determines the structure of
1456:determines the structure of
1329:, very much akin to the way
1154:, and the definition of the
1030:, special classes of rings (
541:{\displaystyle \mathbb {Z} }
29:Ring theory (disambiguation)
3847:Weibel, Charles A. (2013),
3778:Pierce, Richard S. (1982),
3153:{\displaystyle \sigma _{i}}
3011:homogeneous coordinate ring
2466:is the continuous function
2306:. When the equality holds,
1524:Wedderburn's little theorem
1192:unique factorization domain
693:Unique factorization domain
3901:
3574:Judson, Thomas W. (1997),
3550:Cambridge University Press
3498:Cambridge University Press
3279:ascending chain conditions
3066:. The main example is the
2973:
2877:
2866:
2809:One can also consider the
2804:finiteness of class number
2587:is usually denoted by Pic(
2381:be a commutative ring and
2337:
1568:Dimension theory (algebra)
1565:
1348:
1321:over a ring is an abelian
1278:
1209:
1127:
1097:and with the discovery of
453:Tensor product of algebras
26:
3652:10.1007/978-1-4419-8616-0
3614:10.1007/978-1-4612-0525-8
3078:states that this ring is
3015:Hilbert's Nullstellensatz
2283:have the same dimension.
1515:Cartan–Brauer–Hua theorem
1221:Hilbert's Nullstellensatz
1138:if its multiplication is
1072:Hilbert's Nullstellensatz
3812:Rowen, Louis H. (1988),
3463:Rings, Fields and Groups
3363:, Ch I, Definition 2.2.3
3205:William Kingdon Clifford
2997:forms a ring called the
2991:affine algebraic variety
2794:will be taken to be the
2462:. (The rank of a module
1496:Hopkins–Levitzki theorem
1464:Jacobson density theorem
1454:Artin–Wedderburn theorem
731:Formal power series ring
681:Integrally closed domain
3708:Commutative Ring Theory
3429:Emmy Noether: 1882–1935
3399:, Ch I, Corollary 3.8.1
3387:, Ch I, Proposition 3.5
2864:are Morita equivalent.
1762:The Krull dimension of
1303:noncommutative geometry
1289:Noncommutative geometry
1263:by gluing together the
1251:, and augmented with a
1168:Principal ideal domains
1117:Glossary of ring theory
1095:noncommutative geometry
1062:algebraic number theory
740:Algebraic number theory
433:Total ring of fractions
3265:in which they defined
3225:mathematical structure
3219:, were studied within
3181:William Rowan Hamilton
3154:
3127:
3052:
2838:
2749:
2714:
2581:
2543:
2452:
2406:
2271:of a commutative ring
2238:
2210:
2177:
2108:
2080:
2051:
2027:
1948:
1893:
1787:
1752:
1708:
1653:
1543:Skolem–Noether theorem
1486:Zariski–Samuel theorem
1429:Some relevant theorems
1379:linear transformations
1237:Alexander Grothendieck
1188:principal ideal domain
1044:homological properties
897:Noncommutative algebra
880:
833:
796:
748:Algebraic number field
699:Principal ideal domain
588:
542:
480:Frobenius endomorphism
3267:left and right ideals
3248:algebras over a field
3207:was an enthusiast of
3155:
3128:
3072:symmetric polynomials
3053:
2936:matrices over a field
2839:
2802:) is a finite group (
2750:
2715:
2582:
2544:
2453:
2407:
2239:
2211:
2178:
2109:
2081:
2052:
2028:
1949:
1894:
1788:
1753:
1709:
1654:
1403:matrix multiplication
1356:Representation theory
1351:Representation theory
1345:Representation theory
1309:(rings that are also
1076:Fermat's Last Theorem
1048:polynomial identities
881:
834:
797:
589:
543:
3782:Associative Algebras
3514:Faith, Carl (1999),
3137:
3082:
3033:
2895:nilradical of a ring
2817:
2727:
2618:
2553:
2470:
2424:
2385:
2257:transcendence degree
2224:
2190:
2121:
2094:
2061:
2037:
1958:
1919:
1800:
1773:
1726:
1663:
1592:
1490:principal ideal ring
1415:associative algebras
1395:algebraic operations
1367:algebraic structures
1307:associative algebras
1275:Noncommutative rings
1086:Noncommutative rings
1016:algebraic structures
903:Noncommutative rings
852:
814:
777:
621:Non-associative ring
557:
530:
487:Algebraic structures
263:Group with operators
206:Complemented lattice
41:Algebraic structures
3814:Ring Theory, Vol. I
3676:Lam, T. Y. (2003),
3638:Lam, T. Y. (2001),
3600:Lam, T. Y. (1999),
3209:split-biquaternions
3174:hypercomplex number
2911:The concept of the
2886:noncommutative ring
2884:The structure of a
2880:Noncommutative ring
2773:divisor class group
2312:Cohen–Macaulay ring
1907:A commutative ring
1526:states that finite
1448:Structure theorems
1281:Noncommutative ring
1176:Euclidean algorithm
1130:Commutative algebra
1067:commutative algebra
662:Commutative algebra
501:Associative algebra
383:Algebraic structure
317:Composition algebra
77:Quasigroup and loop
3211:, which he called
3150:
3123:
3048:
3021:Ring of invariants
3007:projective variety
2948:endomorphism rings
2869:Algebraic K-theory
2834:
2745:
2710:
2577:
2539:
2448:
2414:projective modules
2402:
2340:Morita equivalence
2334:Morita equivalence
2316:regular local ring
2269:integral extension
2234:
2206:
2173:
2104:
2076:
2047:
2023:
1944:
1901:Hilbert polynomial
1889:
1888:
1846:
1783:
1748:
1704:
1649:
1545:characterizes the
1299:algebraic geometry
1217:Algebraic geometry
1212:Algebraic geometry
1206:Algebraic geometry
1148:algebraic geometry
1111:Ring (mathematics)
1058:Algebraic geometry
916:Semiprimitive ring
876:
829:
792:
600:Related structures
584:
538:
474:Inner automorphism
460:Ring homomorphisms
3860:978-0-8218-9132-2
3585:Kimberling, Clark
3240:Joseph Wedderburn
3221:universal algebra
3126:{\displaystyle R}
3051:{\displaystyle k}
2757:fractional ideals
2669:
2354:Morita equivalent
2294:. In general, if
1831:
1707:{\displaystyle k}
1572:In this section,
1225:algebraic variety
1172:Euclidean domains
1134:A ring is called
1124:Commutative rings
1054:Commutative rings
1000:
999:
957:Geometric algebra
668:Commutative rings
519:Category of rings
376:
375:
34:Branch of algebra
16:(Redirected from
3892:
3871:
3834:
3808:
3785:
3774:
3762:
3728:
3702:
3672:
3634:
3596:
3580:
3570:
3547:
3536:
3510:
3488:
3447:
3445:
3421:
3415:
3409:
3400:
3394:
3388:
3382:
3376:
3370:
3364:
3358:
3352:
3346:
3340:
3334:
3328:
3322:
3316:
3309:
3283:Irving Kaplansky
3263:theory of ideals
3213:algebraic motors
3179:More precisely,
3170:invariant theory
3159:
3157:
3156:
3151:
3149:
3148:
3132:
3130:
3129:
3124:
3119:
3118:
3100:
3099:
3057:
3055:
3054:
3049:
3027:invariant theory
2976:Ring of integers
2952:Klein four-group
2913:Jacobson radical
2906:
2848:(R). Note that K
2843:
2841:
2840:
2835:
2824:
2811:group completion
2796:ring of integers
2782:For example, if
2754:
2752:
2751:
2746:
2719:
2717:
2716:
2711:
2670:
2668:
2651:
2649:
2648:
2636:
2635:
2586:
2584:
2583:
2578:
2567:
2566:
2561:
2548:
2546:
2545:
2540:
2535:
2534:
2522:
2521:
2500:
2499:
2489:
2457:
2455:
2454:
2449:
2438:
2437:
2432:
2411:
2409:
2408:
2403:
2392:
2292:global dimension
2243:
2241:
2240:
2235:
2233:
2232:
2215:
2213:
2212:
2207:
2205:
2204:
2182:
2180:
2179:
2174:
2172:
2171:
2165:
2148:
2147:
2113:
2111:
2110:
2105:
2103:
2102:
2085:
2083:
2082:
2077:
2075:
2071:
2070:
2056:
2054:
2053:
2048:
2046:
2045:
2032:
2030:
2029:
2024:
2022:
2018:
2017:
2007:
2006:
2001:
2000:
1984:
1983:
1978:
1977:
1967:
1966:
1953:
1951:
1950:
1945:
1943:
1939:
1938:
1928:
1927:
1898:
1896:
1895:
1890:
1887:
1886:
1885:
1874:
1873:
1865:
1860:
1859:
1854:
1853:
1845:
1815:
1814:
1813:
1793:-primary ideals.
1792:
1790:
1789:
1784:
1782:
1781:
1757:
1755:
1754:
1749:
1744:
1743:
1713:
1711:
1710:
1705:
1700:
1699:
1681:
1680:
1658:
1656:
1655:
1650:
1648:
1647:
1642:
1641:
1625:
1624:
1619:
1618:
1608:
1607:
1602:
1601:
1473:Goldie's theorem
1458:semisimple rings
1443:Nakayama's lemma
1257:affine varieties
1249:Zariski topology
1200:commutative ring
1184:Euclidean domain
1164:Integral domains
1103:Noetherian rings
1010:is the study of
992:
985:
978:
963:Operator algebra
949:Clifford algebra
885:
883:
882:
877:
872:
871:
859:
838:
836:
835:
830:
828:
827:
822:
801:
799:
798:
793:
791:
790:
785:
763:Ring of integers
757:
754:Integers modulo
705:Euclidean domain
593:
591:
590:
585:
583:
575:
570:
547:
545:
544:
539:
537:
441:Product of rings
427:Fractional ideal
386:
378:
368:
361:
354:
143:Commutative ring
72:Rack and quandle
37:
21:
3900:
3899:
3895:
3894:
3893:
3891:
3890:
3889:
3875:
3874:
3861:
3846:
3824:
3811:
3798:
3788:Springer-Verlag
3777:
3765:
3752:
3742:10.1090/gsm/030
3731:
3718:
3705:
3692:
3682:Springer-Verlag
3675:
3662:
3644:Springer-Verlag
3637:
3624:
3606:Springer-Verlag
3599:
3595:, pp. 3–61
3583:
3573:
3560:
3539:
3526:
3513:
3508:
3491:
3478:
3459:
3456:
3451:
3450:
3443:
3423:
3422:
3418:
3412:Kimberling 1981
3410:
3403:
3395:
3391:
3383:
3379:
3371:
3367:
3359:
3355:
3347:
3343:
3335:
3331:
3323:
3319:
3310:
3306:
3301:
3287:Noetherian ring
3166:
3140:
3135:
3134:
3110:
3091:
3080:
3079:
3031:
3030:
3023:
2999:coordinate ring
2983:
2978:
2972:
2967:
2898:
2882:
2876:
2871:
2855:
2851:
2847:
2815:
2814:
2725:
2724:
2655:
2640:
2627:
2616:
2615:
2556:
2551:
2550:
2513:
2468:
2467:
2427:
2422:
2421:
2383:
2382:
2375:
2352:are said to be
2342:
2336:
2222:
2221:
2188:
2187:
2119:
2118:
2092:
2091:
2064:
2059:
2058:
2035:
2034:
2011:
1994:
1971:
1956:
1955:
1932:
1917:
1916:
1867:
1847:
1804:
1798:
1797:
1771:
1770:
1724:
1723:
1691:
1672:
1661:
1660:
1635:
1612:
1595:
1590:
1589:
1578:Krull dimension
1570:
1564:
1559:
1500:Noetherian ring
1468:primitive rings
1431:
1399:matrix addition
1358:is a branch of
1353:
1347:
1291:
1279:Main articles:
1277:
1233:coordinate ring
1214:
1208:
1196:integral domain
1132:
1126:
1024:representations
996:
967:
966:
899:
889:
888:
863:
850:
849:
817:
812:
811:
780:
775:
774:
755:
725:Polynomial ring
675:Integral domain
664:
654:
653:
555:
554:
528:
527:
513:Involutive ring
398:
387:
381:
372:
343:
342:
341:
312:Non-associative
294:
283:
282:
272:
252:
241:
240:
229:Map of lattices
225:
221:Boolean algebra
216:Heyting algebra
190:
179:
178:
172:
153:Integral domain
117:
106:
105:
99:
53:
35:
32:
23:
22:
15:
12:
11:
5:
3898:
3896:
3888:
3887:
3877:
3876:
3873:
3872:
3859:
3844:
3822:
3809:
3796:
3775:
3763:
3750:
3729:
3716:
3703:
3690:
3673:
3660:
3635:
3622:
3597:
3581:
3571:
3558:
3537:
3524:
3511:
3506:
3489:
3476:
3455:
3452:
3449:
3448:
3441:
3416:
3401:
3389:
3377:
3365:
3353:
3351:, Theorem 31.4
3349:Matsumura 1989
3341:
3339:, Theorem 13.4
3337:Matsumura 1989
3329:
3317:
3303:
3302:
3300:
3297:
3252:Artinian rings
3183:put forth the
3165:
3162:
3147:
3143:
3122:
3117:
3113:
3109:
3106:
3103:
3098:
3094:
3090:
3087:
3047:
3044:
3041:
3038:
3022:
3019:
2982:
2979:
2974:Main article:
2971:
2968:
2966:
2963:
2878:Main article:
2875:
2872:
2853:
2849:
2845:
2833:
2830:
2827:
2823:
2755:is the set of
2744:
2741:
2738:
2735:
2732:
2721:
2720:
2709:
2706:
2703:
2700:
2697:
2694:
2691:
2688:
2685:
2682:
2679:
2676:
2673:
2667:
2664:
2661:
2658:
2654:
2647:
2643:
2639:
2634:
2630:
2626:
2623:
2576:
2573:
2570:
2565:
2560:
2538:
2533:
2528:
2525:
2520:
2516:
2512:
2509:
2506:
2503:
2498:
2492:
2488:
2484:
2481:
2478:
2475:
2447:
2444:
2441:
2436:
2431:
2401:
2398:
2395:
2391:
2374:
2371:
2338:Main article:
2335:
2332:
2231:
2203:
2198:
2195:
2184:
2183:
2170:
2164:
2160:
2157:
2154:
2151:
2146:
2141:
2138:
2135:
2132:
2129:
2126:
2101:
2074:
2069:
2044:
2021:
2016:
2010:
2005:
1999:
1993:
1990:
1987:
1982:
1976:
1970:
1965:
1942:
1937:
1931:
1926:
1911:is said to be
1905:
1904:
1884:
1881:
1878:
1872:
1864:
1858:
1852:
1844:
1841:
1838:
1834:
1830:
1827:
1824:
1821:
1818:
1812:
1807:
1794:
1780:
1767:
1747:
1742:
1737:
1734:
1731:
1718:has dimension
1703:
1698:
1694:
1690:
1687:
1684:
1679:
1675:
1671:
1668:
1646:
1640:
1634:
1631:
1628:
1623:
1617:
1611:
1606:
1600:
1566:Main article:
1563:
1560:
1558:
1555:
1554:
1553:
1535:
1534:
1521:
1519:division rings
1512:
1506:
1492:
1482:
1470:
1460:
1446:
1445:
1440:
1430:
1427:
1385:, and studies
1349:Main article:
1346:
1343:
1317:of modules. A
1276:
1273:
1229:maximal ideals
1210:Main article:
1207:
1204:
1128:Main article:
1125:
1122:
1099:quantum groups
1036:division rings
998:
997:
995:
994:
987:
980:
972:
969:
968:
960:
959:
931:
930:
924:
918:
912:
900:
895:
894:
891:
890:
887:
886:
875:
870:
866:
862:
858:
839:
826:
821:
802:
789:
784:
772:-adic integers
765:
759:
750:
736:
735:
734:
733:
727:
721:
720:
719:
707:
701:
695:
689:
683:
665:
660:
659:
656:
655:
652:
651:
650:
649:
637:
636:
635:
629:
617:
616:
615:
597:
596:
595:
594:
582:
578:
574:
569:
565:
562:
548:
536:
515:
509:
503:
497:
483:
482:
476:
470:
456:
455:
449:
443:
437:
436:
435:
429:
417:
411:
399:
397:Basic concepts
396:
395:
392:
391:
374:
373:
371:
370:
363:
356:
348:
345:
344:
340:
339:
334:
329:
324:
319:
314:
309:
303:
302:
301:
295:
289:
288:
285:
284:
281:
280:
277:Linear algebra
271:
270:
265:
260:
254:
253:
247:
246:
243:
242:
239:
238:
235:Lattice theory
231:
224:
223:
218:
213:
208:
203:
198:
192:
191:
185:
184:
181:
180:
171:
170:
165:
160:
155:
150:
145:
140:
135:
130:
125:
119:
118:
112:
111:
108:
107:
98:
97:
92:
87:
81:
80:
79:
74:
69:
60:
54:
48:
47:
44:
43:
33:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3897:
3886:
3883:
3882:
3880:
3870:
3866:
3862:
3856:
3852:
3851:
3845:
3842:
3841:0-12-599842-2
3838:
3833:
3829:
3825:
3823:0-12-599841-4
3819:
3815:
3810:
3807:
3803:
3799:
3797:0-387-90693-2
3793:
3789:
3784:
3783:
3776:
3773:
3769:
3764:
3761:
3757:
3753:
3751:0-8218-2169-5
3747:
3743:
3739:
3735:
3730:
3727:
3723:
3719:
3717:0-521-36764-6
3713:
3709:
3704:
3701:
3697:
3693:
3691:0-387-00500-5
3687:
3683:
3679:
3674:
3671:
3667:
3663:
3661:0-387-95183-0
3657:
3653:
3649:
3645:
3641:
3636:
3633:
3629:
3625:
3623:0-387-98428-3
3619:
3615:
3611:
3607:
3603:
3598:
3594:
3593:Marcel Dekker
3590:
3586:
3582:
3579:
3578:
3572:
3569:
3565:
3561:
3559:0-521-36086-2
3555:
3551:
3546:
3545:
3538:
3535:
3531:
3527:
3525:0-8218-0993-8
3521:
3517:
3512:
3509:
3507:0-521-27288-2
3503:
3499:
3496:, Cambridge:
3495:
3490:
3487:
3483:
3479:
3477:0-7131-3476-3
3473:
3469:
3465:
3464:
3458:
3457:
3453:
3444:
3442:3-7643-3019-8
3438:
3434:
3430:
3426:
3425:Dick, Auguste
3420:
3417:
3414:, p. 18.
3413:
3408:
3406:
3402:
3398:
3393:
3390:
3386:
3381:
3378:
3374:
3369:
3366:
3362:
3357:
3354:
3350:
3345:
3342:
3338:
3333:
3330:
3326:
3321:
3318:
3314:
3308:
3305:
3298:
3296:
3294:
3293:
3288:
3284:
3280:
3276:
3272:
3268:
3264:
3260:
3255:
3253:
3249:
3245:
3241:
3237:
3232:
3230:
3226:
3222:
3218:
3214:
3210:
3206:
3202:
3201:coquaternions
3198:
3194:
3190:
3189:biquaternions
3186:
3182:
3177:
3175:
3171:
3163:
3161:
3145:
3141:
3115:
3111:
3107:
3104:
3101:
3096:
3092:
3085:
3077:
3073:
3069:
3065:
3061:
3042:
3036:
3028:
3020:
3018:
3016:
3012:
3008:
3004:
3000:
2996:
2992:
2988:
2980:
2977:
2969:
2964:
2962:
2960:
2955:
2953:
2949:
2945:
2941:
2937:
2934:
2930:
2925:
2922:
2918:
2914:
2909:
2905:
2901:
2896:
2891:
2887:
2881:
2873:
2870:
2865:
2863:
2859:
2828:
2812:
2807:
2805:
2801:
2797:
2793:
2789:
2785:
2780:
2778:
2774:
2770:
2766:
2762:
2758:
2739:
2733:
2730:
2707:
2698:
2692:
2689:
2680:
2674:
2671:
2665:
2662:
2656:
2645:
2641:
2632:
2628:
2621:
2614:
2613:
2612:
2610:
2606:
2602:
2598:
2594:
2590:
2571:
2563:
2523:
2518:
2514:
2510:
2507:
2504:
2490:
2479:
2476:
2473:
2465:
2461:
2442:
2434:
2419:
2415:
2396:
2380:
2372:
2370:
2367:
2363:
2359:
2355:
2351:
2347:
2341:
2333:
2331:
2329:
2325:
2321:
2317:
2313:
2309:
2305:
2301:
2297:
2293:
2289:
2284:
2282:
2278:
2274:
2270:
2266:
2262:
2258:
2254:
2250:
2245:
2219:
2196:
2193:
2162:
2158:
2155:
2152:
2149:
2139:
2136:
2133:
2130:
2127:
2124:
2117:
2116:
2115:
2089:
2072:
2019:
2008:
2003:
1991:
1988:
1985:
1980:
1968:
1940:
1929:
1914:
1910:
1902:
1882:
1879:
1876:
1862:
1856:
1842:
1839:
1836:
1832:
1828:
1822:
1816:
1805:
1795:
1768:
1765:
1761:
1760:
1759:
1735:
1732:
1721:
1717:
1714:over a field
1696:
1692:
1688:
1685:
1682:
1677:
1673:
1666:
1644:
1632:
1629:
1626:
1621:
1609:
1604:
1587:
1583:
1579:
1575:
1569:
1561:
1556:
1552:
1548:
1547:automorphisms
1544:
1540:
1539:
1538:
1533:
1529:
1525:
1522:
1520:
1516:
1513:
1510:
1509:Morita theory
1507:
1505:
1504:Artinian ring
1501:
1497:
1493:
1491:
1487:
1483:
1481:
1478:
1474:
1471:
1469:
1465:
1461:
1459:
1455:
1451:
1450:
1449:
1444:
1441:
1439:
1436:
1435:
1434:
1428:
1426:
1424:
1420:
1416:
1412:
1408:
1404:
1400:
1396:
1392:
1388:
1384:
1383:vector spaces
1380:
1376:
1372:
1368:
1365:
1361:
1357:
1352:
1344:
1342:
1340:
1336:
1332:
1328:
1327:endomorphisms
1324:
1320:
1316:
1312:
1311:vector spaces
1308:
1304:
1300:
1296:
1290:
1286:
1282:
1274:
1272:
1270:
1266:
1262:
1258:
1254:
1250:
1246:
1242:
1238:
1234:
1230:
1226:
1222:
1218:
1213:
1205:
1203:
1201:
1197:
1193:
1189:
1185:
1181:
1177:
1173:
1169:
1165:
1161:
1160:prime numbers
1157:
1153:
1149:
1145:
1141:
1137:
1131:
1123:
1121:
1119:
1118:
1113:
1112:
1106:
1104:
1100:
1096:
1092:
1087:
1083:
1081:
1077:
1073:
1069:
1068:
1063:
1059:
1055:
1051:
1049:
1045:
1041:
1037:
1033:
1029:
1025:
1021:
1017:
1013:
1009:
1005:
993:
988:
986:
981:
979:
974:
973:
971:
970:
965:
964:
958:
954:
953:
952:
951:
950:
945:
944:
943:
938:
937:
936:
929:
925:
923:
919:
917:
913:
911:
910:Division ring
907:
906:
905:
904:
898:
893:
892:
864:
848:
846:
840:
824:
810:
809:-adic numbers
808:
803:
787:
773:
771:
766:
764:
760:
758:
751:
749:
745:
744:
743:
742:
741:
732:
728:
726:
722:
718:
714:
713:
712:
708:
706:
702:
700:
696:
694:
690:
688:
684:
682:
678:
677:
676:
672:
671:
670:
669:
663:
658:
657:
648:
644:
643:
642:
638:
634:
630:
628:
624:
623:
622:
618:
614:
610:
609:
608:
604:
603:
602:
601:
576:
572:
563:
560:
553:
552:Terminal ring
549:
526:
522:
521:
520:
516:
514:
510:
508:
504:
502:
498:
496:
492:
491:
490:
489:
488:
481:
477:
475:
471:
469:
465:
464:
463:
462:
461:
454:
450:
448:
444:
442:
438:
434:
430:
428:
424:
423:
422:
421:Quotient ring
418:
416:
412:
410:
406:
405:
404:
403:
394:
393:
390:
385:→ Ring theory
384:
379:
369:
364:
362:
357:
355:
350:
349:
347:
346:
338:
335:
333:
330:
328:
325:
323:
320:
318:
315:
313:
310:
308:
305:
304:
300:
297:
296:
292:
287:
286:
279:
278:
274:
273:
269:
266:
264:
261:
259:
256:
255:
250:
245:
244:
237:
236:
232:
230:
227:
226:
222:
219:
217:
214:
212:
209:
207:
204:
202:
199:
197:
194:
193:
188:
183:
182:
177:
176:
169:
166:
164:
163:Division ring
161:
159:
156:
154:
151:
149:
146:
144:
141:
139:
136:
134:
131:
129:
126:
124:
121:
120:
115:
110:
109:
104:
103:
96:
93:
91:
88:
86:
85:Abelian group
83:
82:
78:
75:
73:
70:
68:
64:
61:
59:
56:
55:
51:
46:
45:
42:
38:
30:
19:
3849:
3813:
3781:
3771:
3733:
3707:
3677:
3639:
3601:
3588:
3576:
3543:
3515:
3493:
3462:
3428:
3419:
3392:
3380:
3368:
3356:
3344:
3332:
3320:
3307:
3290:
3277:, analyzing
3274:
3259:Emmy Noether
3256:
3236:matrix rings
3233:
3217:Lie algebras
3212:
3193:James Cockle
3178:
3167:
3063:
3059:
3024:
3002:
2994:
2986:
2984:
2965:Applications
2956:
2932:
2928:
2926:
2917:annihilators
2910:
2903:
2899:
2883:
2861:
2857:
2808:
2799:
2791:
2787:
2783:
2781:
2776:
2764:
2760:
2722:
2608:
2604:
2600:
2596:
2593:Picard group
2588:
2463:
2459:
2417:
2378:
2376:
2361:
2357:
2349:
2345:
2343:
2323:
2319:
2310:is called a
2307:
2303:
2299:
2295:
2285:
2280:
2276:
2272:
2264:
2260:
2252:
2248:
2246:
2185:
2087:
1908:
1906:
1763:
1719:
1715:
1585:
1581:
1573:
1571:
1551:simple rings
1536:
1480:Goldie rings
1447:
1432:
1419:Lie algebras
1397:in terms of
1371:representing
1370:
1354:
1339:monoid rings
1292:
1215:
1135:
1133:
1115:
1109:
1107:
1084:
1065:
1052:
1007:
1001:
961:
947:
946:
942:Free algebra
940:
939:
933:
932:
901:
844:
806:
769:
738:
737:
717:Finite field
666:
613:Finite field
599:
598:
525:Initial ring
485:
484:
458:
457:
400:
388:
337:Hopf algebra
275:
268:Vector space
233:
174:
173:
102:Group theory
100:
65: /
3885:Ring theory
3446:, p. 44–45.
3397:Weibel 2013
3385:Weibel 2013
3373:Weibel 2013
3361:Weibel 2013
3242:(1908) and
3229:direct sums
3185:quaternions
2959:quaternions
2420:; let also
2328:homological
1360:mathematics
1180:polynomials
1156:prime ideal
1140:commutative
1136:commutative
1032:group rings
1008:ring theory
922:Simple ring
633:Jordan ring
507:Graded ring
389:Ring theory
322:Lie algebra
307:Associative
211:Total order
201:Semilattice
175:Ring theory
18:Ring Theory
3454:References
3433:Birkhäuser
3292:Noetherian
3244:Emil Artin
3197:tessarines
3195:presented
2867:See also:
2344:Two rings
1315:categories
1227:, and the
1080:arithmetic
928:Commutator
687:GCD domain
3257:In 1920,
3142:σ
3112:σ
3105:…
3093:σ
2734:
2705:→
2693:
2687:→
2675:
2660:↦
2653:→
2646:∗
2638:→
2633:∗
2625:→
2515:⊗
2508:
2502:↦
2483:→
2477:
2197:
2156:
2140:
2128:
1992:⊊
1989:⋯
1986:⊊
1930:⊂
1840:≥
1833:⨁
1817:
1686:⋯
1633:⊊
1630:⋯
1627:⊊
1610:⊊
1502:to be an
1477:semiprime
1407:algebraic
1091:functions
1089:rings of
869:∞
647:Semifield
332:Bialgebra
138:Near-ring
95:Lie group
63:Semigroup
3879:Category
3468:xxvi+383
3427:(1981),
3005:. For a
2940:geometry
2366:category
2330:notion.
2073:′
2020:′
1941:′
1913:catenary
1433:General
1393:and the
1391:matrices
1375:elements
1364:abstract
1335:matrices
1295:matrices
1261:manifold
1245:spectrum
1144:integers
1020:integers
641:Semiring
627:Lie ring
409:Subrings
168:Lie ring
133:Semiring
3869:3076731
3832:0940245
3806:0674652
3760:1811901
3726:1011461
3700:2003255
3670:1838439
3632:1653294
3568:1020298
3534:1657671
3486:1144518
3164:History
2944:physics
2852:(R) = K
2769:regular
2275:, then
2216:is the
1528:domains
1387:modules
1241:schemes
1231:of its
1028:modules
1004:algebra
843:PrĂĽfer
445:•
299:Algebra
291:Algebra
196:Lattice
187:Lattice
3867:
3857:
3839:
3830:
3820:
3804:
3794:
3758:
3748:
3724:
3714:
3698:
3688:
3668:
3658:
3630:
3620:
3566:
3556:
3532:
3522:
3504:
3484:
3474:
3439:
3203:; and
3133:where
2989:is an
2921:simple
2890:simple
2723:where
2267:is an
2218:height
2186:where
1537:Other
1532:fields
1411:groups
1373:their
1331:fields
1319:module
1287:, and
1267:of an
1265:charts
1152:ideals
495:Module
468:Kernel
327:Graded
258:Module
249:Module
148:Domain
67:Monoid
3299:Notes
3269:in a
2767:is a
2763:. If
2599:. If
2416:over
2288:depth
2263:. If
2057:and
1323:group
1269:atlas
1253:sheaf
1012:rings
847:-ring
711:Field
607:Field
415:Ideal
402:Rings
293:-like
251:-like
189:-like
158:Field
116:-like
90:Magma
58:Group
52:-like
50:Group
3855:ISBN
3837:ISBN
3818:ISBN
3792:ISBN
3746:ISBN
3712:ISBN
3686:ISBN
3656:ISBN
3618:ISBN
3554:ISBN
3520:ISBN
3502:ISBN
3472:ISBN
3437:ISBN
3313:rngs
3271:ring
3199:and
3187:and
2931:-by-
2731:Cart
2672:Cart
2474:Spec
2377:Let
2314:. A
2290:and
2279:and
1541:The
1530:are
1494:The
1484:The
1462:The
1452:The
1417:and
1401:and
1060:and
1046:and
123:Ring
114:Ring
3738:doi
3648:doi
3610:doi
3238:by
3062:on
3001:of
2985:If
2919:of
2813:of
2775:of
2759:of
2690:Pic
2607:of
2595:of
2549:.)
2505:dim
2247:If
2220:of
2153:dim
2125:dim
1580:of
1549:of
1381:of
1377:as
1369:by
1002:In
128:Rng
3881::
3865:MR
3863:,
3828:MR
3826:,
3802:MR
3800:,
3790:,
3770:,
3756:MR
3754:,
3744:,
3722:MR
3720:,
3696:MR
3694:,
3684:,
3666:MR
3664:,
3654:,
3646:,
3628:MR
3626:,
3616:,
3608:,
3591:,
3564:MR
3562:,
3552:,
3530:MR
3528:,
3500:,
3482:MR
3480:,
3470:,
3435:,
3404:^
3295:.
3254:.
3191:;
3070::
2961:.
2954:.
2942:,
2902:Ă—
2860:,
2779:.
2348:,
2244:.
2194:ht
2137:ht
2114:,
1903:).
1806:gr
1758::
1413:,
1341:.
1283:,
1271:.
1202:.
1198:⊂
1194:⊂
1190:⊂
1186:⊂
1162:.
1120:.
1105:.
1050:.
1038:,
1034:,
1006:,
955:•
926:•
920:•
914:•
908:•
841:•
804:•
767:•
761:•
752:•
746:•
729:•
723:•
715:•
709:•
703:•
697:•
691:•
685:•
679:•
673:•
645:•
639:•
631:•
625:•
619:•
611:•
605:•
550:•
523:•
517:•
511:•
505:•
499:•
493:•
478:•
472:•
466:•
451:•
439:•
431:•
425:•
419:•
413:•
407:•
3843:.
3740::
3650::
3612::
3327:.
3315:.
3146:i
3121:]
3116:n
3108:,
3102:,
3097:1
3089:[
3086:R
3064:V
3060:G
3046:]
3043:V
3040:[
3037:k
3003:X
2995:X
2987:X
2933:n
2929:n
2904:n
2900:n
2862:S
2858:R
2854:0
2850:0
2846:0
2832:)
2829:R
2826:(
2822:P
2800:R
2792:R
2788:R
2784:R
2777:R
2765:R
2761:R
2743:)
2740:R
2737:(
2708:1
2702:)
2699:R
2696:(
2684:)
2681:R
2678:(
2666:R
2663:f
2657:f
2642:F
2629:R
2622:1
2609:R
2605:F
2601:R
2597:R
2589:R
2575:)
2572:R
2569:(
2564:1
2559:P
2537:)
2532:p
2527:(
2524:k
2519:R
2511:M
2497:p
2491:,
2487:Z
2480:R
2464:M
2460:n
2446:)
2443:R
2440:(
2435:n
2430:P
2418:R
2400:)
2397:R
2394:(
2390:P
2379:R
2362:S
2358:R
2350:S
2346:R
2324:R
2320:R
2308:R
2304:R
2300:R
2296:R
2281:R
2277:S
2273:R
2265:S
2261:k
2253:k
2249:R
2230:p
2202:p
2169:p
2163:/
2159:R
2150:+
2145:p
2134:=
2131:R
2100:p
2088:R
2068:p
2043:p
2015:p
2009:=
2004:n
1998:p
1981:0
1975:p
1969:=
1964:p
1936:p
1925:p
1909:R
1883:1
1880:+
1877:k
1871:m
1863:/
1857:k
1851:m
1843:0
1837:k
1829:=
1826:)
1823:R
1820:(
1811:m
1779:m
1766:.
1764:R
1746:)
1741:m
1736:,
1733:R
1730:(
1720:n
1716:k
1702:]
1697:n
1693:t
1689:,
1683:,
1678:1
1674:t
1670:[
1667:k
1645:n
1639:p
1622:1
1616:p
1605:0
1599:p
1586:n
1582:R
1574:R
1014:—
991:e
984:t
977:v
874:)
865:p
861:(
857:Z
845:p
825:p
820:Q
807:p
788:p
783:Z
770:p
756:n
581:Z
577:1
573:/
568:Z
564:=
561:0
535:Z
367:e
360:t
353:v
31:.
20:)
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