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25:
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of any algebraic structure where it exists, like it was described for examples above. But its existence and, if it exists, the property to be an
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984:
108:
89:
61:
891:, then zero morphisms exist and the situation is not different from non-unital structures considered in the previous section.
46:
314:
These objects are described jointly not only based on the common singleton and trivial group structure, but also because of
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126:
75:
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35:
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If an algebraic structure requires the multiplicative identity, but neither its preservation by morphisms nor
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510:
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In categories where the multiplicative identity must be preserved by morphisms, but can equal to zero, the
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This article is about trivial or zero algebraic structures. For zero elements in algebraic structures, see
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847:
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object cannot exist because it may contain only one element. In particular, the zero ring is not a
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The zero object, also by definition, must be an initial object, which means that a morphism
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For structures requiring the multiplication structure inside the zero object, such as the
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object can exist. But not as initial object because identity-preserving morphisms from
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which has both an additive and multiplicative identity is trivial if and only if
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24:
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The zero object, by definition, must be a terminal object, which means that a
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is, in the sense explained below, the simplest object of such structure. As a
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1004:
850:, this abstract and somewhat mysterious mathematical object is not a field.
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Instances of the zero object include, but are not limited to the following:
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175:. The aforementioned abelian group structure is usually identified as
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depend on exact definition of the multiplicative identity; see
18:
368:, because there are no non-zero elements. This structure is
899:
Zero vector spaces and zero modules are usually denoted by
681:
The zero ring, zero module and zero vector space are the
671:
to obtain 2-dimensional zero vector (leftmost). Rules of
325:
by an element of the base ring (or field) is defined as:
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is an especially ubiquitous example of a zero object, a
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must exist and be unique for an arbitrary object
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must exist and be unique for an arbitrary object
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Any trivial algebra is also a trivial ring. A trivial
697:. However, the zero ring is not a zero object in the
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401:
624:{\displaystyle {\begin{bmatrix}\,\\\,\end{bmatrix}}}
49:. Unsourced material may be challenged and removed.
623:
581:{\displaystyle {\begin{bmatrix}0\\0\end{bmatrix}}}
580:
437:
907:). This is always the case when they occur in an
469:is simultaneously a zero vector space considered
275:is also used, although it may be ambiguous, as a
349:The most general of them, the zero module, is a
183:, so the object itself is typically denoted as
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8:
768:, and hence its image is isomorphic to
663:Element of the zero space, written as empty
634:
846:. If mathematicians sometimes talk about a
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384:, since this equality implies that for all
199:to any other (under a unique isomorphism).
823:sense) depend on exact definition of the
723:. This morphism maps any element of
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607:
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597:
552:
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400:
125:. For the zero object in a category, see
109:Learn how and when to remove this message
16:Algebraic structure with only one element
791:) in each module (or vector space)
438:{\displaystyle r=r\times 1=r\times 0=0.}
488:. A trivial algebra is an example of a
772:. For modules and vector spaces, this
667:(rightmost one), is multiplied by 2Ă0
458:
448:In this case it is possible to define
316:shared category-theoretical properties
7:
705:of the zero ring in any other ring.
484:The trivial ring is an example of a
47:adding citations to reliable sources
865:do not exist. For example, in the
827: 1 in a specified structure.
14:
481:is simultaneously a zero module.
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179:, and the only element is called
880:is the initial object, not
764:in vector spaces. This map is a
195:) since every trivial object is
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191:trivial object (of a specified
34:needs additional citations for
364:, there is only one possible,
1:
304:zero-dimensional vector space
783:is the only empty-generated
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321:In the last three cases the
171:structure, which is also an
127:Initial and terminal objects
753:, to the zero element
749:, the only element of
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505:over a field with an empty
139:to and from the zero object
1072:
969:Cambridge University Press
830:If the definition of
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58:"Zero object" algebra
1056:Objects (category theory)
931:Examples of vector spaces
695:category of vector spaces
662:
351:finitely-generated module
926:Triviality (mathematics)
687:category of pseudo-rings
459:§ Unital structures
965:Rings and factorization
825:multiplicative identity
936:Field with one element
848:field with one element
685:of, respectively, the
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283:with a trivial action.
187:. One often refers to
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963:David Sharpe (1987).
745:. This morphism maps
673:matrix multiplication
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583:
495:The zero-dimensional
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323:scalar multiplication
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1015:Barile, Margherita.
921:Nildimensional space
861:to any object where
821:category-theoretical
701:, since there is no
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549:
467:algebra over a field
399:
242:algebra over a field
43:improve this article
691:category of modules
513:zero. It is also a
509:. It therefore has
246:algebra over a ring
153:algebraic structure
996:Barile, Margherita
951:List of zero terms
787:(or 0-dimensional
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486:rng of square zero
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799:Unital structures
703:ring homomorphism
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1000:"Trivial Module"
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1017:"Zero Module"
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665:column vector
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515:trivial group
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99:February 2012
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60: â
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55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
1020:
1003:
977:trivial ring
976:
964:
946:Zero element
903:(instead of
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886:
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872:the ring of
869:
852:
829:
816:
807:object is a
802:
779:
766:monomorphism
756:
737:
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707:
683:zero objects
680:
669:empty matrix
522:
503:vector space
498:vector space
494:
490:zero algebra
483:
477:, a trivial
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362:trivial ring
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288:vector space
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235:trivial ring
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123:Zero element
105:
96:
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65:
53:
41:Please help
36:verification
33:
1041:Ring theory
838:, then the
817:zero object
762:zero vector
374:commutative
370:associative
271:. The term
269:zero module
163:, and as a
151:of a given
149:zero object
1051:0 (number)
1035:Categories
971:. p.
532:Properties
308:zero space
212:zero group
197:isomorphic
69:newspapers
1022:MathWorld
1005:MathWorld
785:submodule
635: âč0
511:dimension
473:. Over a
424:×
412:×
376:. A ring
366:0 Ă 0 = 0
230:zero ring
161:singleton
137:Morphisms
975: :
915:See also
895:Notation
874:integers
727:to
710:morphism
693:and the
521:, and a
519:addition
353:with an
335:, where
306:or just
290:(over a
281:G-module
259:(over a
193:category
177:addition
159:it is a
819:in the
543:
479:algebra
462:below.
388:within
298:), the
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169:trivial
145:algebra
83:scholar
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863:1 â 0
844:field
836:1 â 0
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517:over
507:basis
471:below
382:1 = 0
355:empty
292:field
286:As a
279:is a
255:As a
223:As a
208:group
206:As a
165:magma
90:JSTOR
76:books
981:ISBN
870:Ring
803:The
755:0 â
372:and
261:ring
225:ring
181:zero
62:news
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244:or
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