Finitely projective modules with respect to a semidualizing module

2019 ◽  
Vol 18 (03) ◽  
pp. 1950049
Author(s):  
Lixin Mao
Keyword(s):  
Commutative Ring ◽  
Projective Modules ◽  
Artinian Rings ◽  
Semidualizing Module ◽  
Coherent Rings

Let [Formula: see text] be a commutative ring. We define and study [Formula: see text]-projective modules with respect to a semidualizing [Formula: see text]-module [Formula: see text], which are called [Formula: see text]–[Formula: see text]-projective modules. As consequences, we characterize several rings such as [Formula: see text]-coherent rings and Artinian rings using [Formula: see text]–[Formula: see text]-projective modules. Some known results are extended.

Envelopes, covers and semidualizing modules

2019 ◽  
Vol 18 (07) ◽  
pp. 1950137
Author(s):  
Lixin Mao
Keyword(s):  
Commutative Ring ◽  
Noetherian Rings ◽  
Artinian Rings ◽  
Injective Modules ◽  
Semidualizing Modules ◽  
Coherent Rings ◽  
The Relationship

Given an [Formula: see text]-module [Formula: see text] and a class of [Formula: see text]-modules [Formula: see text] over a commutative ring [Formula: see text], we investigate the relationship between the existence of [Formula: see text]-envelopes (respectively, [Formula: see text]-covers) and the existence of [Formula: see text]-envelopes or [Formula: see text]-envelopes (respectively, [Formula: see text]-covers or [Formula: see text]-covers) of modules. As a consequence, we characterize coherent rings, Noetherian rings, perfect rings and Artinian rings in terms of envelopes and covers by [Formula: see text]-projective, [Formula: see text]-flat, [Formula: see text]-injective and [Formula: see text]-[Formula: see text]-injective modules, where [Formula: see text] is a semidualizing [Formula: see text]-module.

scholarly journals Artinian bimodule with quasi-Frobenius bimodule of translations

2019 ◽  
Vol 29 (2) ◽  
pp. 103-119
Author(s):  
Aleksandr A. Nechaev ◽  
Vadim N. Tsypyschev
Keyword(s):  
Commutative Ring ◽  
Recurrent Sequence ◽  
Artinian Rings ◽  
Linear Recurrent Sequence ◽  
Left And Right ◽  
Additional Assumptions

Abstract The possibility to generalize the notion of a linear recurrent sequence (LRS) over a commutative ring to the case of a LRS over a non-commutative ring is discussed. In this context, an arbitrary bimodule AMB over left- and right-Artinian rings A and B, respectively, is associated with the equivalent bimodule of translations CMZ, where C is the multiplicative ring of the bimodule AMB and Z is its center, and the relation between the quasi-Frobenius conditions for the bimodules AMB and CMZ is studied. It is demonstrated that, in the general case, the fact that AMB is a quasi-Frobenius bimodule does not imply the validity of the quasi-Frobenius condition for the bimodule CMZ. However, under some additional assumptions it can be shown that if CMZ is a quasi-Frobenius bimodule, then the bimodule AMB is quasi-Frobenius as well.

On commutative rings with uniserial dimension

2014 ◽  
Vol 14 (01) ◽  
pp. 1550008 ◽  
Author(s):  
A. Ghorbani ◽  
Z. Nazemian
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Artinian Rings ◽  
Uniform Dimension ◽  
Uniserial Modules

In this paper, we define and study a valuation dimension for commutative rings. The valuation dimension is a measure of how far a commutative ring deviates from being valuation. It is shown that a ring R with valuation dimension has finite uniform dimension. We prove that a ring R is Noetherian (respectively, Artinian) if and only if the ring R × R has (respectively, finite) valuation dimension if and only if R has (respectively, finite) valuation dimension and all cyclic uniserial modules are Noetherian (respectively, Artinian). We show that the class of all rings of finite valuation dimension strictly lies between the class of Artinian rings and the class of semi-perfect rings.

scholarly journals Rank Element of a Projective Module

1965 ◽  
Vol 25 ◽  
pp. 113-120 ◽  
Author(s):  
Akira Hattori
Keyword(s):  
Finite Group ◽  
Commutative Ring ◽  
Local Ring ◽  
Projective Module ◽  
Projective Modules ◽  
Local Theorem ◽  
Complete Local Ring ◽  
A Finite Group

In § 1 of this note we first define the trace of an endomorphism of a projective module P over a non-commutative ring A. Then we call the trace of the identity the rank element r(P) of P, which we shall illustrate by several examples. For a projective module P over the groupalgebra of a finite group G, the rank element of P is essentially the character of G in P. In § 2 we prove that under certain assumption two projective modules Pi and P2 over an algebra over a complete local ring o are isomorphic if and only if their rank elements are identical. This is a type of proposition asserting that two representations are equivalent if and only if their characters are identical, and in fact, when A is the groupalgebra, the above theorem may be considered as another formulation of Swan’s local theorem [9]).

scholarly journals AC-GORENSTEIN RINGS AND THEIR STABLE MODULE CATEGORIES

2018 ◽  
Vol 107 (02) ◽  
pp. 181-198
Author(s):  
JAMES GILLESPIE
Keyword(s):  
The Other ◽  
Homotopy Category ◽  
Gorenstein Ring ◽  
Model Structure ◽  
Projective Modules ◽  
Stable Module Category ◽  
Stable Module ◽  
Coherent Rings ◽  
Quillen Equivalent ◽  
Compactly Generated

We introduce what is meant by an AC-Gorenstein ring. It is a generalized notion of Gorenstein ring that is compatible with the Gorenstein AC-injective and Gorenstein AC-projective modules of Bravo–Gillespie–Hovey. It is also compatible with the notion of $n$ -coherent rings introduced by Bravo–Perez. So a $0$ -coherent AC-Gorenstein ring is precisely a usual Gorenstein ring in the sense of Iwanaga, while a $1$ -coherent AC-Gorenstein ring is precisely a Ding–Chen ring. We show that any AC-Gorenstein ring admits a stable module category that is compactly generated and is the homotopy category of two Quillen equivalent abelian model category structures. One is projective with cofibrant objects that are Gorenstein AC-projective modules while the other is an injective model structure with fibrant objects that are Gorenstein AC-injectives.

Classification of rings with toroidal co-annihilating-ideal graphs

2020 ◽  
pp. 2150006
Author(s):  
Rana Khoeilar ◽  
Jafar Amjadi
Keyword(s):  
Commutative Ring ◽  
Artinian Rings ◽  
Vertex Set

Let [Formula: see text] be a commutative ring with identity. The co-annihilating-ideal graph of [Formula: see text], denoted by [Formula: see text], is a graph whose vertex set is the set of all nonzero proper ideals of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent whenever [Formula: see text]. In this paper, we study the planarity and genus of [Formula: see text]. In particular, we characterize all Artinian rings [Formula: see text] for which the genus of [Formula: see text] is zero or one.

Homological Dimensions with Respect to a Semidualizing Module and Tensor Products of Algebras

2015 ◽  
Vol 22 (02) ◽  
pp. 215-222
Author(s):  
Maryam Salimi ◽  
Elham Tavasoli ◽  
Siamak Yassemi
Keyword(s):  
Commutative Ring ◽  
Noetherian Ring ◽  
Tensor Products ◽  
Projective Dimension ◽  
Injective Dimension ◽  
Semidualizing Module ◽  
Homological Dimensions

Let C be a semidualizing module for a commutative ring R. It is shown that the [Formula: see text]-injective dimension has the ability to detect the regularity of R as well as the [Formula: see text]-projective dimension. It is proved that if D is dualizing for a Noetherian ring R such that id R(D) = n < ∞, then [Formula: see text] for every flat R-module F. This extends the result due to Enochs and Jenda. Finally, over a Noetherian ring R, it is shown that if M is a pure submodule of an R-module N, then [Formula: see text]. This generalizes the result of Enochs and Holm.

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