Multiplicative sets, left strongly prime ideals, and left strongly prime radicals in rings

Abstract

Left commutative multiplicative sets {ie397-01} for an associative ring R are defined. In particular, this notion includes commutative multiplicative sets of the associative ring. We also define the notion of a left {ie397-02}-ideal and prove that each left {ie397-03}-ideal, maximal with respect to being disjoint from {ie397-04}, is left strongly prime. Using a technique developed for insulators for a left ideal, we also characterize the left strongly prime radical of a left ideal of the ring R, which was known only for two-sided ideals.