Some results on skew generalized power series rings
A partially ordered set \((S,\leq)\) is said to be Artinian if every strictly decreasing sequence of elements of \(S\) is finite and is said to be narrow if every subset of pairwise order-incomparable elements of \(S\) is finite. Let \(R\) be a ring and \((S,\leq)\) a strictly ordered monoid, both of them are not necessarily commutative. Let \(\omega: S\longrightarrow\mathrm{End }E\) a monoid homomorphism. For \(s\in S\), put \(\omega_s=\omega(s)\). Denote by \(RS,\omega\) the set of all the functions \(f:S\longrightarrow R\) such that the support \(\mathrm{supp}(f)=\{s\in S;f(s)\not=0\}\) is Artinian and narrow. Then for any \(s\in S\) and \(f,g\in RS,\omega\), the set \(X_s(f,g)=\{(x,y)\in \mathrm{supp}(f)\times\mathrm{supp}(g); s=xy\}\) is finite. Thus one can define the product \(fg:S\longrightarrow R\) as follows \(\displaystyle fg(s)=\sum_{(u,v)\in X_s(f,g)}f(u)\omega_ug(v)\). With pointwise addition and this multiplication, \(RS,\omega\) becomes a ring, called the ring of skew generalized power series with coefficients in \(R\) and exponents in \(S\). This kind of construction includes many classical ring constructions. In this paper, the authors study when \(RS,\omega\) has a (flat) projective socle and when it is local, semilocal, semiperfect, semiregular, left quasi-duo, clean, exchange, right stable range one, projective-free and I-ring.
scientific article; zbMATH DE number 6693688