Seminear-rings characterized by their \({\mathcal S}\)-ideals. I
A (right) seminearring is an algebra \(R=(R,+,\cdot)\) such that \((R,+)\) and \((R,\cdot)\) are semigroups connected by \((a+b)c=ac+bc\) for all \(a, b, c \in R\). It is further assumed that \(R\) has an absorbing zero 0, defined by \(a+0=0+a=a\) and \(a0=0a=0\) for all \(a \in R\). A subsemigroup \(I\) of \((R,+)\) is called a right [left] \(\mathcal S\)-ideal of \(R\) if \(xr \in I\) [\(rx \in I]\) holds for all \(x \in I\) and \(r \in R\), and an \({\mathcal S}\)-ideal if both is the case. Suppressing the unexplained indication ``\(\mathcal S\)'' in the following, let \(A\) and \(B\) be ideals of \(R\). Then \(AB=\{\sum^n_{k=1} a_k b_k \mid a_k \in A,\;b_k \in B\}\) need not be an ideal of \(R\). A seminearring \(R\) is called strongly idempotent (briefly: \(SI\)) if \(I=\langle I^2\rangle\) holds for each ideal \(I\) of \(R\), where \(\langle I^2\rangle\) denotes the ideal generated by \(I^2\), a condition equivalent to \(A \cap B=\langle AB\rangle\) for all ideals of \(R\). The paper contains several statements on \(SI\)-seminearrings and examples for \(SI\)-seminearrings which are neither regular nor simple.
scientific article; zbMATH DE number 851718