276:. That condition, called the ascending chain condition on principal ideals or ACCP, is strictly weaker than the BFD condition, and strictly stronger than the atomic condition (in other words, even if there exist infinite chains of proper divisors, it can still be that every
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is precisely a ring in which an analogue of the fundamental theorem of arithmetic holds, this question is readily answered. However, one notices that there are two aspects of the fundamental theorem of the arithmetic: first, that any integer is the finite product of
125:). Therefore, it is also natural to ask under what conditions particular elements of a ring can be "decomposed" without requiring uniqueness. The concept of an atomic domain addresses this.
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The ring of integers (that is, the set of integers with the natural operations of addition and multiplication) satisfy many important properties. One such property is the
71:
168:. Such a product is allowed to involve the same irreducible element more than once as a factor.) Any such expression is called a factorization of
59:
in that this decomposition of an element into irreducibles need not be unique; stated differently, an irreducible element is not necessarily a
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there is no bound on the number of irreducible factors. If on the contrary the number of factors is bounded for every non-zero non-unit
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divisors). Every unique factorization domain obviously satisfies these two conditions, but neither implies unique factorization.
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can be viewed as merely an abstract set in which one can perform the operations of addition and multiplication; analogous to the
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A. Grams, Atomic rings and the ascending chain condition for principal ideals. Proc. Cambridge Philos. Soc. 75 (1974), 321–329.
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with at least one irreducible factor for each step of the chain), so there cannot be any infinite strictly ascending chain of
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D. D. Anderson, D. F. Anderson, M. Zafrullah, Factorization in integral domains; J. Pure and
Applied Algebra 69 (1990) 1–19
112:. Thus, when considering abstract rings, a natural question to ask is under what conditions such a theorem holds. Since a
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to 1 can exceed this bound in length (since the quotient at every step can be factored, producing a factorization of
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Important examples of atomic domains include the class of all unique factorization domains and all
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Two independent conditions that are both strictly stronger than the BFD condition are the
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In an atomic domain, it is possible that different factorizations of the same element
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121:, and second, that this product is unique up to rearrangement (and multiplication by
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P.M. Cohn, Bezout rings and their subrings; Proc. Camb. Phil.Soc. 64 (1968) 251–264
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is referred to as an atomic domain. (The product is necessarily finite, since
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have different lengths. It is even possible that among the factorizations of
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is claimed to hold in Cohn's paper, this is known to be false.
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can be written in at least one way as a finite product of
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If such a bound exists, no chain of proper divisors from
70:. More generally, any integral domain satisfying the
72:ascending chain condition on principal ideals
8:
204:); formally this means that for each such
74:(ACCP) is an atomic domain. Although the
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291:: any two factorizations of any given
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55:. Atomic domains are different from
280:possesses a finite factorization).
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110:fundamental theorem of arithmetic
89:of an integral domain an "atom".
355:Bezout rings and their subrings
307:has but a finite number of non-
152:can be written as a product of
295:have the same length) and the
1:
198:bounded factorization domain
81:The term "atomic" is due to
57:unique factorization domains
18:Bounded factorization domain
297:finite factorization domain
114:unique factorization domain
387:
208:there exists an integer
47:in which every non-zero
285:half-factorial domain
154:irreducible elements
141:. If every non-zero
53:irreducible elements
41:factorization domain
31:, more specifically
371:Commutative algebra
164:are not defined in
97:In this section, a
87:irreducible element
68:Noetherian domains
240:with none of the
162:infinite products
16:(Redirected from
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270:principal ideals
249:invertible then
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85:, who called an
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45:integral domain
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119:prime numbers
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61:prime element
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37:atomic domain
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353:P.M. Cohn,
299:condition (
287:condition (
166:ring theory
33:ring theory
29:mathematics
315:References
129:Definition
93:Motivation
83:P. M. Cohn
309:associate
365:Category
143:non-unit
103:integers
76:converse
49:non-unit
357:, 1968.
192:, then
303:: any
137:be an
43:is an
253:<
196:is a
123:units
35:, an
133:Let
99:ring
301:FFD
289:HFD
272:of
230:...
202:BFD
148:of
39:or
27:In
367::
257:.
217:=
172:.
156:,
105:.
63:.
305:x
293:x
278:x
274:R
266:x
262:x
255:N
251:n
246:i
242:x
236:n
232:x
228:2
225:x
222:1
219:x
215:x
210:N
206:x
200:(
194:R
190:x
186:x
182:x
170:x
158:R
150:R
146:x
135:R
20:)
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