A characterization of $S_1$-projective modules
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Recently, Zhao, Pu, Chen, and Xiao introduced and investigated novel concepts regarding $S$-torsion exact sequences, $S$-torsion commutative diagrams, and $S_i$-projective modules (for $i = 1, 2$) in the context of a commutative ring $R$ and a multiplicative subset $S$ of $R$. Their research included various results, such as proving that an $R$-module is $S_1$-projective if it is $S$-torsion isomorphic to a projective module. In this paper, we further examine properties of $S$-torsion exact sequences and $S$-torsion commutative diagrams, and we establish the equivalence between an $R$-module being $S_1$-projective and its $S$-torsion isomorphism to a projective module.
Abstract
Recently, Zhao, Pu, Chen, and Xiao introduced and investigated novel concepts regarding $S$-torsion exact sequences, $S$-torsion commutative diagrams, and $S_i$-projective modules (for $i = 1, 2$) in the context of a commutative ring $R$ and a multiplicative subset $S$ of $R$. Their research included various results, such as proving that an $R$-module is $S_1$-projective if it is $S$-torsion isomorphic to a projective module. In this paper, we further examine properties of $S$-torsion exact sequences and $S$-torsion commutative diagrams, and we establish the equivalence between an $R$-module being $S_1$-projective and its $S$-torsion isomorphism to a projective module.