Uniformly S-Projective Modules and Uniformly S-Projective Uniformly S-Covers
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Recently, Zhang and Qi introduced and studied the concept of uniformly S projective (u-S-projective) modules, where S represents a multiplicative subset of a ring. In this paper, we first derive a u-S version of the well-known result that a module is projective if and only if it is a direct summand of a free module. We then provide a characterization of u-S-projective modules in terms of projective modules, specifically when S is regular,and then extend the projective basis lemma to the u-S setting. Finally, we show that a u-S-projective u-S-cover is characterized by a u-S-superfluous submodule, analogous to the way a projective cover is characterized by a superfluous submodule.
Abstract
Recently, Zhang and Qi introduced and studied the concept of uniformly S projective (u-S-projective) modules, where S represents a multiplicative subset of a ring. In this paper, we first derive a u-S version of the well-known result that a module is projective if and only if it is a direct summand of a free module. We then provide a characterization of u-S-projective modules in terms of projective modules, specifically when S is regular,and then extend the projective basis lemma to the u-S setting. Finally, we show that a u-S-projective u-S-cover is characterized by a u-S-superfluous submodule, analogous to the way a projective cover is characterized by a superfluous submodule.
참고문헌 (6)
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- F. Wang, "Foundations of Commutative Rings and Their Modules", Algebr. Appl., 22 (2016), Springer
- X. L. Zhang, "The u-S-weak global dimension of commutative rings", Commun. Korean Math. Soc., 38(1), pp.97~112 (2023)
- X. L. Zhang, "Characterizing S-projective modules and S-semisimple rings by uniformity", J. Commut. Algebra, 15(1), pp.139~149 (2023)
- W. Zhao, "On ϕ-von Neumann regular rings", J. Korean Math. Soc, 50(1), pp.219~229 (2013)