Abstract

A right \(R\)-module \(M_R\) is called a CS module (or an extending module) if every submodule of \(M\) is essential in a direct summand of \(M\). A ring \(R\) is called right CS if \(R_R\) is a CS module. This paper deals with rings for which all finitely generated right ideals are CS. One result is: For a semiprime right Goldie ring \(R\) with right uniform dimension at least 2, it is shown that every finitely generated right ideal of \(R\) is CS if and only if \(R\) is a ring direct sum \(R=R_1\oplus\cdots\oplus R_n\), where each \(R_i\) is either a right Ore domain, or a prime right and left Goldie, right and left semihereditary ring. Consequently, the following fact can be proved as a corollary in this paper: For a simple right CS right QI ring \(R\) with right uniform dimension at least 2, Boyle’s conjecture that every simple right QI-ring is hereditary is equivalent to the statement that all finitely generated right (or left) ideals of \(R\) are CS.