An Equational Characterization of Quasi-injective Modules

1997 ◽  
Vol 25 (11) ◽  
pp. 3545-3549
Author(s):  
A. Laradji
Keyword(s):  
Injective Modules ◽  
Equational Characterization

Quasi-pseudo Principally Injective Modules

2009 ◽  
Vol 16 (03) ◽  
pp. 397-402 ◽  
Author(s):  
Avanish Kumar Chaturvedi ◽  
B. M. Pandeya ◽  
A. J. Gupta
Keyword(s):  
Injective Module ◽  
Injective Modules

In this paper, the concept of quasi-pseudo principally injective modules is introduced and a characterization of commutative semi-simple rings is given in terms of quasi-pseudo principally injective modules. An example of pseudo M-p-injective module which is not M-pseudo injective is given.

scholarly journals Acyclic Complexes and Gorenstein Rings

2020 ◽  
Vol 27 (03) ◽  
pp. 575-586
Author(s):  
Sergio Estrada ◽  
Alina Iacob ◽  
Holly Zolt
Keyword(s):  
Analogous Result ◽  
Gorenstein Ring ◽  
Gorenstein Rings ◽  
Injective Modules ◽  
Flat Modules ◽  
Coherent Ring ◽  
Acyclic Complexes ◽  
Gorenstein Flat ◽  
Gorenstein Injective

For a given class of modules [Formula: see text], let [Formula: see text] be the class of exact complexes having all cycles in [Formula: see text], and dw([Formula: see text]) the class of complexes with all components in [Formula: see text]. Denote by [Formula: see text][Formula: see text] the class of Gorenstein injective R-modules. We prove that the following are equivalent over any ring R: every exact complex of injective modules is totally acyclic; every exact complex of Gorenstein injective modules is in [Formula: see text]; every complex in dw([Formula: see text][Formula: see text]) is dg-Gorenstein injective. The analogous result for complexes of flat and Gorenstein flat modules also holds over arbitrary rings. If the ring is n-perfect for some integer n ≥ 0, the three equivalent statements for flat and Gorenstein flat modules are equivalent with their counterparts for projective and projectively coresolved Gorenstein flat modules. We also prove the following characterization of Gorenstein rings. Let R be a commutative coherent ring; then the following are equivalent: (1) every exact complex of FP-injective modules has all its cycles Ding injective modules; (2) every exact complex of flat modules is F-totally acyclic, and every R-module M such that M+ is Gorenstein flat is Ding injective; (3) every exact complex of injectives has all its cycles Ding injective modules and every R-module M such that M+ is Gorenstein flat is Ding injective. If R has finite Krull dimension, statements (1)–(3) are equivalent to (4) R is a Gorenstein ring (in the sense of Iwanaga).

ESSENTIALLY SLIGHTLY COMPRESSIBLE MODULES AND RINGS

2012 ◽  
Vol 05 (02) ◽  
pp. 1250028 ◽  
Author(s):  
Abhay K. Singh
Keyword(s):  
Noetherian Ring ◽  
Slightly Compressible ◽  
Injective Modules ◽  
Nonzero Homomorphism

In this paper, the concepts of essentially slightly compressible modules and essentially slightly compressible rings are introduced, and related properties are investigated. The notion of essentially slightly compressible modules and rings are generalization of essentially compressible modules and rings introduced by [P. F. Smith and M. R. Vedadi, Essentially compressible modules and rings, J. Algebra304 (2006) 812–831]. I have also provided the characterization of such modules in terms of nonsingular injective modules. It has been shown that over a noetherian ring for a uniform module, essentially slightly compressible modules, slightly compressible modules and nonzero homomorphism from M into U for a nonzero uniform submodule U of M are equivalent. Throughout this paper, all rings are associative and all modules are unital right R-modules.

scholarly journals New characterization of $\Sigma $-injective modules

2008 ◽  
Vol 136 (10) ◽  
pp. 3461-3466 ◽  
Author(s):  
K. I. Beidar ◽  
S. K. Jain ◽  
Ashish K. Srivastava
Keyword(s):  
Injective Modules

scholarly journals The Powerdomain of Indexed Valuations

2002 ◽  
Vol 9 (38) ◽  
Author(s):  
Daniele Varacca
Keyword(s):  
Denotational Semantics ◽  
Point Of View ◽  
Domain Theory ◽  
Theoretic Point ◽  
Equational Characterization ◽  
Imperative Language

This paper is about combining nondeterminism and probabilities. We study this phenomenon from a domain theoretic point of view. In domain theory, nondeterminism is modeled using the notion of powerdomain, while probability is modeled using the powerdomain of valuations. Those two functors do not combine well, as they are. We define the notion of powerdomain of indexed valuations, which can be combined nicely with the usual nondeterministic powerdomain. We show an equational characterization of our construction. Finally we discuss the computational meaning of indexed valuations, and we show how they can be used, by giving a denotational semantics of a simple imperative language.

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