Abstract. A ring R is called left max-coherent provided that every maximal left ideal is finitely presented. M I (resp. M F ) denotes the class of all max-injective left R-modules (resp. all max-flat right R-modules). We prove, in this article, that over a left max-coherent ring every right R-module has an MF -preenvelope, and every left R-module has an MIcover. Furthermore, it is shown that a ring R is left max-injective if and only if any left R-module has an epic MI -cover if and only if any right R-module has a monic MF -preenvelope. We also give several equivalent characterizations of M I-injectivity and M I-flatness. Finally, MIdimensions of modules and rings are studied in terms of max-injective modules with the left derived functors of Hom.
A right ideal I of a ring R is small in case for every proper right ideal K of R, K + I = R. A right R-module M is called P S-injective if every R-homomorphism f : aR → M for every principally small right ideal aR can be extended to R → M . A ring R is called right P S-injective if R is P S-injective as a right R-module. We develop, in this article, P S-injectivity as a generalization of P -injectivity and small injectivity. Many characterizations of right P S-injective rings are studied. In light of these facts, we get several new properties of a right GP F ring and a semiprimitive ring in terms of right P S-injectivity. Related examples are given as well.
Let R be a ring and let H be a subgroup of a finite group G. We consider the weak global dimension, cotorsion dimension and weak Gorenstein global dimension of the skew group ring RσG and its coefficient ring R. Under the assumption that RσG is a separable extension over RσH, it is shown that RσG and RσH share the same homological dimensions. Several known results are then obtained as corollaries. Moreover, we investigate the relationships between the homological dimensions of RσG and the homological dimensions of a commutative ring R, using the trivial RσG-module.
Abstract. Let R be a ring and N il * (R) be the prime radical of R. In this paper, we say that a ring R is left N il * -coherent if N il * (R) is coherent as a left R-module. The concept is introduced as the generalization of left J-coherent rings and semiprime rings. Some properties of N il * -coherent rings are also studied in terms of N -injective modules and N -flat modules.
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