On quasi-ideals in contracted semigroup rings
Let A be a ring with identity and S a semigroup with a zero 0 and an identity e. Then the contracted semigroup ring \(A_ 0[S]\) of S over A is the free unitary left A-module with \(S'=S\setminus \{0\}\) as a basis and a multiplication determined by that of S. Identifying A with the subring \(Ae=\{ae|\) \(a\in A\}\) of \(A_ 0[S]\), the latter is the ring- theoretical product \(<AS>=\{\sum a_ is_ i|\) \(a_ i\in A\), \(s_ i\in S\}\) of its subsets A and S. In this notation, each pair of quasi- ideals \(A_ 1\) of A and \(S_ 1\) of S determines a quasi-ideal \(Q=<A_ 1S_ 1>\). The latter are called formal quasi-ideals of \(A_ 0[S]\) and determine the corresponding \(A_ 1\) and \(S_ 1\) uniquely if \(Q\neq \{0\}\) holds. These formal quasi-ideals of \(A_ 0[S]\) are used to obtain several results on arbitrary quasi-ideals Q of S, in particular on minimal ones. For instance, each quasi-ideal Q of \(A_ 0[S]\) is contained in a unique smallest formal quasi-ideal, and if a minimal quasi-ideal Q of \(A_ 0[S]\) contains an element \(a\neq 0\) of A [s\(\neq 0\) of S], then Q is a formal quasi-ideal and \(S=G^ 0\) a 0-group [A a division ring].
scientific article; zbMATH DE number 3935341
scientific article