# On the existence of nonzero injective covers and projective envelopes of modules

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- Properties of RD-projective and RD-injective modules. Lixin Mao // Turkish Journal of Mathematics;2011, Vol. 35 Issue 2, p187
In this paper, we first study RD-projective and RD-injective modules using, among other things, covers and envelopes. Some new characterizations for them are obtained. Then we introduce the RD-projective and RD-injective dimensions for modules and rings. The relations between the RD-homological...

- Rings Whose Simple Singular Modules are PS-Injective. YUEMING XIANG; LUNQUN OUYANG // Kyungpook Mathematical Journal;Sep2014, Vol. 54 Issue 3, p471
Let R be a ring. A right R-module M is PS-injective if every R-homomorphism f : aR â†’ M for every principally small right ideal aR can be extended to R â†’ M. We investigate, in this paper, rings whose simple singular modules are PS-injective. New characterizations of semiprimitive...

- RELATIVE HOMOLOGICAL COALGEBRAS. Enochs, E. E.; Lopez-Ramos, J. A. // Acta Mathematica Hungarica;2004, Vol. 104 Issue 4, p331
We study classes of relative injective and projective comodules and extend well-known results about projective comodules given in [7]. The existence of covers and envelopes by these classes of comodules is also studied and used to characterize the projective dimension of a coalgebra. We also...

- MODULES THAT HAVE A SUPPLEMENT IN EVERY COATOMIC EXTENSION. TÜRKMEN, BURCU NİŞANCI // Miskolc Mathematical Notes;Sep2015, Vol. 16 Issue 1, p543
Let R be a ring and M be an R-module. M is said to be an E*-module (respectively, an EE*-module) if M has a supplement (respectively, ample supplements) in every coatomic extension N, i.e. N/M is coatomic. We prove that if a module M is an EE*-module, every submodule of M is an E*-module, and...

- Nilpotent elements and reduced rings. Junchao Wei; Libin Li // Turkish Journal of Mathematics;2011, Vol. 35 Issue 2, p341
In this paper, we show the following results: (1) R is a min-leftsemicentral ring if and only if eR(1 - e)Re = 0 for alle ? MEl(R) ; (2) Quasi-normal rings, NI rings and weakly reversible rings are all minleftsemicentral ring; (3) R is left MC2 ring if and only if aRe = 0 implies eRa = 0 for...

- Gorenstein projective dimensions of complexes. Liu, Zhong; Zhang, Chun // Acta Mathematica Sinica;Jul2011, Vol. 27 Issue 7, p1395
We show that over a right coherent left perfect ring R, a complex C of left R-modules is Gorenstein projective if and only if C is Gorenstein projective in R-Mod for all m âˆˆ â„¤. Basing on this we show that if R is a right coherent left perfect ring then Gpd( C) = sup{Gpd( C)| m âˆˆ...

- In a Right Perfect Right Self-Injective Ring Right PF? Nguyen Thi Bach Kim // Vietnam Journal of Mathematics;Dec2000, Vol. 28 Issue 4, p321
Examines the formulations of right perfect right self-injective ring R. Sufficient conditions for a semiperfect ring; Questions for a right perfect right self-injective ring R; Characterization of a right pseudo-frobenius ring.

- STRONGLY NONCOSINGULAR MODULES. ALAGÖZ, Y.; DURĞUN, Y. // Bulletin of the Iranian Mathematical Society;Aug2016, Vol. 42 Issue 4, p999
An R-module M is called strongly noncosingular if it has no nonzero Rad-small (cosingular) homomorphic image in the sense of Harada. It is proven that (1) an R-module M is strongly noncosingular if and only if M is coatomic and noncosingular; (2) a right perfect ring R is Artinian hereditary...

- On modules which satisfy the radical formula. SARAÇ, Bülent; TIRAŞ, Yücel // Turkish Journal of Mathematics;Mar2013, Vol. 37 Issue 2, p195
In this paper, the authors prove that every representable module over a commutative ring with identity satisfies the radical formula. With this result, they extend the class of modules satisfying the radical formula from that of Artinian modules to a larger one. They conclude their work by...