Abstract

Left permutable multiplicative sets S for an associative ring R are defined. Particularly, this notion includes commutative multiplicative sets of the associative ring. We also define the notion of the left S-ideal and prove, that each left S-ideal, maximal with respect to being disjoint from S, is left strongly prime.

Highlights

  • All considered rings are associative with identity element

  • We recall that a left ideal p ⊂ R is called strongly prime if for each u ∈ p there exists a finite set α1, ..., αn ∈ R, n = n(u), such that rα1u, ..., rαnu ∈ p, where r ∈ R, implies that r ∈ p

  • Multiplicative sets and left strongly prime ideals Recall that a subset S of a ring R is multiplicative, if it is multiplicatively closed and contains the identity element of R

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Summary

Introduction

All considered rings are associative with identity element. A ⊂ B means that A is a proper subset of B. We recall that a left ideal p ⊂ R is called (left) strongly prime if for each u ∈ p there exists a finite set α1, ..., αn ∈ R, n = n(u), such that rα1u, ..., rαnu ∈ p, where r ∈ R, implies that r ∈ p. Basic properties of the left strongly prime ideals are considered in [3, 4, 7]. 1. Multiplicative sets and left strongly prime ideals Recall that a subset S of a ring R is multiplicative, if it is multiplicatively closed and contains the identity element of R.

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