Journal article
Irreducible elements in commutative rings with Zero-Divisors, II
Houston Journal of Mathematics, Vol.39(3), pp.741-752
2013
Abstract
Let R be a commutative ring with identity. For a, b ε R, a and b are associates (resp., strong associates, very strong associates) denoted a ∼ b (resp., a ≈ b, a ≅ b) if (a) = (b) (resp., a = ub for some unit u ε R, a ∼ b and either a = b = 0 or a ≠ 0 and a = rb implies r is a unit). A nonunit a ε R is irreducible (resp., strongly irreducible, very strongly irreducible) if a = bc (b, c ε R) implies a ∼ b or a ∼ c (resp., a ≈ b or a ≈ c, a ≅ b or a ≅ c) and a is m-irreducible if (a) ⊆ (b) ⊆ R implies (b) = (a) or (b) = R. The ring R is said to be atomic (resp., strongly atomic, very strongly atomic, m-atomic) if each nonzero nonunit of R is a finite product of irreducible (resp., strongly irreducible, very strongly irreducible, m-irreducible) elements. In this paper we collect the known various characterizations of the different types of irreducible elements and give a number of new ones. We also continue the investigation of the various forms of atomicity. © 2013 University of Houston.
Details
- Title: Subtitle
- Irreducible elements in commutative rings with Zero-Divisors, II
- Creators
- S. ChunD.D. Anderson
- Resource Type
- Journal article
- Publication Details
- Houston Journal of Mathematics, Vol.39(3), pp.741-752
- ISSN
- 0362-1588
- Language
- English
- Date published
- 2013
- Academic Unit
- Mathematics
- Record Identifier
- 9984230626502771
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