ON REGULAR $QB$-IDEALS

Abstract

Let $I$ be a regular ideal of a ring $R$. It is shown that $I$ is a $QB$-ideal if and only if for all finitely generated projective right $R$-module $A$ with $AI=A$, if $B_1$ and $B_2$ are any right $R$-modules such that $A \oplus B_1 \cong A \oplus B_2$, then there exists a pair of orthogonal ideals $I_1$ and $I_2$ and $B_1 \oplus C_1 \cong B_2 \oplus C_2$ such that $C_1 I_1 \cong C_1$ and $CI_2 \cong C_2$.