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Factorization in commutative rings with zero divisors. (English) Zbl 0865.13001
The aim of the paper is to generalize to commutative rings with zero divisors, $$R$$, some known facts about factorization into irreducible elements and uniqueness properties which arise for commutative integral domains. Moreover, the paper unifies some of these results which have already been done. In section 2 of the paper under review, several concepts of irreducibility are given in such a way that each one of them have more requirements than the former. These definitions provide their corresponding forms of atomicity which are considered in section 3. Section 4 is devoted to introduce and study the concept of $$(\alpha,\beta)$$-unique factorization ring where $$\alpha$$ ranges over the different forms of atomicity and $$\beta$$ over the forms of factorization as products of the different forms of irreducibility. Finally, the authors give several examples in section 5 and study the extension of the irreducibility and factorization to the polynomial ring $$R[X]$$ and the power series ring $$R[[X]]$$.

MSC:
 13A05 Divisibility and factorizations in commutative rings 13F15 Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial)
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References:
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