Let R be a nonsingular ring with essential right socle S and (R) over bar = R/S. We show that, for a projective right R-module P, a finite direct sum decomposition of P modulo its socle can be lifted to P. Let pd((R) over bar)((M) over bar) denote the projective dimension of a right (R) over bar -module (M) over bar. We have the equivalent conditions: pd(R)((M) over bar) = pd((R) over bar)((M) over bar) if (M) over bar is not (R) over bar -projective; and a submodule P of a free R-module is projective only if P modulo its socle is (R) over bar -projective. [References: 8]
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