Right chain domains with prescribed value holoid
Throughout, all rings have an identity and are associative, but are not necessarily commutative. A ring \(R\) is called a right chain domain if \(R\) has no zero-divisors and for any \(a,b\in R\) either \(aR\subseteq bR\) or \(bR\subsetneq aR\). A ring \(R\) is called right invariant if \(Ra\subseteq aR\) for all \(a\in R\). It is clear that the nonzero principal right ideals of \(R\) form a semigroup \(H(R)\) (the value semigroup of \(R\)) under multiplication if and only if \(R\) is right invariant. A semigroup \(H\) with identity \(e\) and a (total) order relation \(\leq\) is called an ordered semigroup if \(a\leq b\) implies \(ac\leq bc\), \(ca\leq cb\) for elements \(a,b,c\in H\). If, in addition, \(a<b\) holds if and only if \(b=ac\) for \(e\neq c\), then \(H\) is called a holoid. In this paper, the authors get the following important result: Let \(G_i\), \(i\in I\), be a set of ordered groups \(G_i\) for a (totally) ordered index set \(I\) and let \(N_i\) be a convex subset of \(G_i\) which contains the positive cone \(G^+_i\) of \(G_i\) for each \(i\). Then there exists a right invariant right chain domain \(R\) such that the associated value semigroup \(H(R)\) is isomorphic to the split holoid \(H=H_{i\in I}(G_i,N_i)\) of a family of ordered groups \(G_i\). The ring \(R\) is obtained as a suitable subring of the generalized power series ring over the wreath product of the \(G_i\). The rings \(R\) constructed in the above result are local right Bezout domains.
scientific article; zbMATH DE number 815914