Weak dimension and right distributivity of skew generalized power series rings. (English) Zbl 1218.16037

Mazurek, Ryszard; Ziembowski, Michał

J. Math. Soc. Japan 62, No. 4, 1093-1112 (2010).

Let \(S\) be a multiplicative monoid (i.e. a semigroup with identity) and let \(\leq\) be an order relation on the set \(S\). The monoid \(S\) is said to be strictly ordered if for all \(s,t,v\in S\) the inequality \(s<t\) implies \(sv<tv\) and \(vs<vt\). An ordered monoid is positively ordered if \(s\geq 1\) for every \(s\in S\).
Let \(R\) be an associative ring with identity and let \(\omega\colon S\to\text{End}(R)\) be a monoid homomorphism. Then the authors define a skew generalized power series ring \(R[[S,\omega]]\) as the set of all maps \(f\colon S\to R\) whose support \(\text{Supp}(f)=\{s\in S\mid f(s)\neq 0\}\) is Artinian and narrow (i.e. it does not contain infinite strictly decreasing chains of elements and it does not contain infinite subsets of pairwise order-incomparable elements). Hence, the skew series ring \(R[[S,\sigma]]\) can be considered as a special case of a skew generalized power series ring \(R[[S,\omega]]\).
The aim of this paper is to study relationships between the weak dimension of a skew generalized power rings \(R[[S,\omega]]\) with positively strictly ordered monoid \(S\) and the properties of the lattice of right ideals of \(R[[S,\omega]]\). When \(S\) is a positively strictly ordered monoid, then the authors give sufficient and necessary conditions under which \(R[[S,\omega]]\) has weak dimension less than or equal to one. In particular, for such an \(S\) they show that \(R[[S,\omega]]\) is distributive and \(\omega(s)\) is injective for every \(s\in S\). These results generalize the main results of R. Mazurek and M. Ziembowski [Publ. Mat., Barc. 53, No. 2, 257-271 (2009; Zbl 1176.16034)].

Reviewer: S. V. Mihovski (Plovdiv)

MSC:

16W60	Valuations, completions, formal power series and related constructions (associative rings and algebras)
16D25	Ideals in associative algebras
16E10	Homological dimension in associative algebras
16D40	Free, projective, and flat modules and ideals in associative algebras
16D50	Injective modules, self-injective associative rings
16E50	von Neumann regular rings and generalizations (associative algebraic aspects)

Keywords:

skew generalized power series rings; weak dimension; right distributive rings; right Bézout rings; lattices of right ideals

Citations:

Zbl 1176.16034

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