Proper classes generated by simple modules

Let R be a ring with unity. A short exact sequence E of left R-modules is said to be neat-exact if every simple left R-module is projective with respect to it. We call it P-pure-exact if for every left primitive ideal P of R, the sequence obtained by taking the tensor product of E from the left by R/P is exact. These give proper classes of short exact sequences of left R-modules. The characterization of N-domains, that is, the commutative domains such that neatness and P-purity coincide, has been given recently by László Fuchs: they are the commutative domains where every maximal ideal is projective (and so necessarily finitely generated in the commutative domain case). We extend this sufficient condition to commutative rings using the Auslander- Bridger tranpose of simple R-modules, that is, we prove that if R is a commutative ring where every maximal ideal is projective and finitely generated, then neatness and P-purity coincide. Conversely, we show that the necessary condition holds for commutative rings with zero socle, that is, we show that if R is a commutative ring where neatness and P-purity coincide and if R has zero socle, then every maximal ideal of the ring R is projective and finitely generated. Keywords: Neat short exact sequence, P-pure short exact sequence, simple Rmodule, the Auslander-Bridger transpose, left primitive ideal, proper class, N-domain,commutative rings with zero socle, maximal ideal, projective R-module, injective Rmodule, ?at R-module.