Journal article
Irreducible elements in commutative rings with zero-divisors
Houston Journal of Mathematics, Vol.37(3), pp.741-744
2011
Abstract
Let R be a commutative ring with zero-divisors. A nonunit element a ∈ R is irreducible if a = bc implies (a) = (b) or (a) = (c). We show that if a, b ∈ R with a irreducible and (a) ⊆ (b) ⊆ R, then a is a zero-divisor and b is a non zero-divisor. It follows that a ∈ R is irreducible if and only if (1) (a) is maximal in the set of proper principal ideals of R or (2) (a) is maximal in the set of principal ideals generated by zero-divisors. Thus a chain (a1) ⊆ • • • ⊆ (an) of principal ideals generated by irreducible elements must have n ≤2. © 2011 University of Houston.
Details
- Title: Subtitle
- Irreducible elements in commutative rings with zero-divisors
- Creators
- D.D. AndersonS. Chun
- Resource Type
- Journal article
- Publication Details
- Houston Journal of Mathematics, Vol.37(3), pp.741-744
- ISSN
- 0362-1588
- Language
- English
- Date published
- 2011
- Academic Unit
- Mathematics
- Record Identifier
- 9984230421902771
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