Rings whose nonsingular modules have projective covers

Abstract

We determine rings $R$ with the property that all (finitely generated) nonsingular right $R$-modules have projective covers. These are just the rings with $t$-supplemented (finitely generated) free right modules. Hence, they are called right (finitely) $\Sigma -t$-supplemented. It is also shown that a ring $R$ for which every cyclic nonsingular right $R$-module has a projective cover is exactly a right $t$-supplemented ring. It is proved that, for a continuous ring $R$, the property of right $\Sigma -t$-supplementedness is equivalent to the semisimplicity of $R/Z_2(R_R)$, while the property of being right finitely $\Sigma -t$-supplemented is equivalent to the right self-injectivity of $R/Z_2(R_R)$. Moreover, for a von Neumann regular ring $R/Z_2(R_R)$, the properties of being right $\Sigma -t$-supplemented, right finitely \Sigma -t-supplemented, and right t-supplemented are equivalent to the semisimplicity, right self-injectivity, and right continuity of $R/Z_2(R_R)$, respectively.