Generalized power series with a limited number of factorizations
The paper deals with generalized power series ring. Let's first review it. Let \((S,+,\leq)\) be a strictly ordered commutative monoid, that is if \(s,t,u\in S\) are such that \(s<t\) then \(s+u<t+u\). A subset \(F\) of \(S\) is said Artinian if each decreasing sequence of elements of \(F\) stabilizes and is narrow if it contains no infinite set of incomparable elements. Now, let \(R\) be a commutative ring with unit. The generalized power series ring \([[R,S^{\leq}]]\) consists of all the formal sums \(\displaystyle f=\sum_{s\in S}a_sX^s\) with \(a_s\in A\) such that \(\mathrm{supp}(f)=\{s\in S; a_s\not=0\}\) is Artinian and narrow. This type of rings include (Laurent) power series and large (Laurent) polynomials. In the third section of this paper, the authors investigate when \([[R,S^{\leq}]]\) is an integral domain where every nonzero nonunit has only finitely many divisors up to associates. As it is very known, beside the context of integral domains, factorization theory is defined for commutative rings with zero divisors, where, as expected many difficulties and problems arise. In the fourth section, the authors investigate when \([[R,S^{\leq}]]\) satisfies the ``limited factorization'' properties. Many applications, examples and diagrams are given.
scientific article; zbMATH DE number 7634468
scientific article