Clean coalgebras and clean comodules of finitely generated projective modules

Let R be a commutative ring with multiplicative identity and P is a finitely generated projective R-module. If P∗ is the set of R-module homomorphism from P to R, then the tensor product P∗⊗RP can be considered as an R-coalgebra. Furthermore, P and P∗ is a comodule over coalgebra P∗⊗RP. Using the Morita context, this paper give sufficient conditions of clean coalgebra P∗⊗RP and clean P∗⊗RP-comodule P and P∗. These sufficient conditions are determined by the conditions of module P and ring R.

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Inertial subalgebras of algebras possessing finite automorphism groups

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700012295 ◽

1979 ◽

Vol 28 (3) ◽

pp. 335-345 ◽

Cited By ~ 1

Author(s):

Nicholas S. Ford

Keyword(s):

Finite Group ◽

Commutative Ring ◽

Jacobson Radical ◽

Automorphism Groups ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Finitely Generated ◽

Fixed Ring ◽

A Finite Group ◽

Necessary And Sufficient

AbstractLet R be a commutative ring with identity, and let A be a finitely generated R-algebra with Jacobson radical N and center C. An R-inertial subalgebra of A is a R-separable subalgebra B with the property that B+N=A. Suppose A is separable over C and possesses a finite group G of R-automorphisms whose restriction to C is faithful with fixed ring R. If R is an inertial subalgebra of C, necessary and sufficient conditions for the existence of an R-inertial subalgebra of A are found when the order of G is a unit in R. Under these conditions, an R-inertial subalgebra B of A is characterized as being the fixed subring of a group of R-automorphisms of A. Moreover, A ⋍ B ⊗R C. Analogous results are obtained when C has an R-inertial subalgebra S ⊃ R.

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On Φ-Dedekind, Φ-Prüfer and Φ-Bezout modules

Georgian Mathematical Journal ◽

10.1515/gmj-2017-0049 ◽

2020 ◽

Vol 27 (1) ◽

pp. 103-110

Author(s):

Shahram Motmaen ◽

Ahmad Yousefian Darani

Keyword(s):

Commutative Ring ◽

Principal Ideal ◽

Finitely Generated ◽

Module Homomorphism ◽

Prime Submodule ◽

Bezout Module

AbstractIn this paper, we introduce some classes of R-modules that are closely related to the classes of Prüfer, Dedekind and Bezout modules. Let R be a commutative ring with identity and set\mathbb{H}=\bigl{\{}M\mid M\text{ is an }R\text{-module and }\mathrm{Nil}(M)% \text{ is a divided prime submodule of }M\bigr{\}}.For an R-module {M\in\mathbb{H}}, set {T=(R\setminus Z(R))\cap(R\setminus Z(M))}, {\mathfrak{T}(M)=T^{-1}M} and {P=(\mathrm{Nil}(M):_{R}M)}. In this case, the mapping {\Phi:\mathfrak{T}(M)\to M_{P}} given by {\Phi(x/s)=x/s} is an R-module homomorphism. The restriction of Φ to M is also an R-module homomorphism from M into {M_{P}} given by {\Phi(x)=x/1} for every {x\in M}. A nonnil submodule N of M is said to be Φ-invertible if {\Phi(N)} is an invertible submodule of {\Phi(M)}. Moreover, M is called a Φ-Prüfer module if every finitely generated nonnil submodule of M is Φ-invertible. If every nonnil submodule of M is Φ-invertible, then we say that M is a Φ-Dedekind module. Furthermore, M is said to be a Φ-Bezout module if {\Phi(N)} is a principal ideal of {\Phi(M)} for every finitely generated submodule N of the R-module M. The paper is devoted to the study of the properties of Φ-Prüfer, Φ-Dedekind and Φ-Bezout R-modules.

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PROJECTIVE MODULES AND DIVISOR HOMOMORPHISMS

Journal of Algebra and Its Applications ◽

10.1142/s0219498803000593 ◽

2003 ◽

Vol 02 (04) ◽

pp. 435-449 ◽

Cited By ~ 17

Author(s):

ALBERTO FACCHINI ◽

FRANZ HALTER-KOCH

Keyword(s):

Commutative Ring ◽

Projective Modules ◽

Finitely Generated ◽

Commutative Monoids ◽

Isomorphism Classes

We study some applications of the theory of commutative monoids to the monoid [Formula: see text] of all isomorphism classes of finitely generated projective right modules over a (not necessarily commutative) ring R.

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Rank and dimension functions

Electronic Journal of Linear Algebra ◽

10.13001/1081-3810.2999 ◽

2015 ◽

Vol 29 ◽

pp. 144-155

Author(s):

K. Prasad ◽

Nupur Nandini ◽

Divya Shenoy

Keyword(s):

Vector Space ◽

Partial Order ◽

Commutative Ring ◽

Free Module ◽

Projective Modules ◽

Finitely Generated ◽

The Matrix ◽

Dimension Functions ◽

Rank Of A Matrix ◽

Minus Partial Order

In this paper, we invoke theory of generalized inverses and minus partial order on regular matrices over a commutative ring to define rankâfunction for regular matrices and dimensionâfunction for finitely generated projective modules which are direct summands of a free module. Some properties held by the rank of a matrix and the dimension of a vector space over a field are generalized. Also, a generalization of rank-nullity theorem has been established when the matrix given is regular.

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QF-3 endomorphism rings of Σ-quasi-projective modules

Glasgow Mathematical Journal ◽

10.1017/s0017089500007254 ◽

1988 ◽

Vol 30 (2) ◽

pp. 215-220 ◽

Cited By ~ 1

Author(s):

José L. Gómez Pardo ◽

Nieves Rodríguez González

Keyword(s):

Endomorphism Ring ◽

Projective Module ◽

Sufficient Conditions ◽

Torsion Theory ◽

Projective Modules ◽

Finitely Generated ◽

Endomorphism Rings ◽

Quotient Module ◽

Qf Ring ◽

Necessary And Sufficient

A ring R is called left QF-3 if it has a minimal faithful left R-module. The endomorphism ring of such a module has been recently studied in [7], where conditions are given for it to be a left PF ring or a QF ring. The object of the present paper is to study, more generally, when the endomorphism ring of a Σ-quasi-projective module over any ring R is left QF-3. Necessary and sufficient conditions for this to happen are given in Theorem 2. An useful concept in this investigation is that of a QF-3 module which has been introduced in [11]. If M is a finitely generated quasi-projective module and σ[M] denotes the category of all modules isomorphic to submodules of modules generated by M, then we show that End(RM) is a left QF-3 ring if and only if the quotient module of M modulo its torsion submodule (in the torsion theory of σ[M] canonically defined by M) is a QF-3 module (Corollary 4). Finally, we apply these results to the study of the endomorphism ring of a minimal faithful R-module over a left QF-3 ring, extending some of the results of [7].

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Finite Exchange Property for Quasi-projective Modules

Algebra Colloquium ◽

10.1142/s1005386711000381 ◽

2011 ◽

Vol 18 (03) ◽

pp. 507-518

Author(s):

Huanyin Chen

Keyword(s):

Projective Module ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Exchange Property ◽

Projective Modules ◽

Finitely Generated ◽

Exchange Rings ◽

Necessary And Sufficient ◽

Finitely Generated Modules

In this paper, we obtain several necessary and sufficient conditions under which a quasi-projective module has the finite exchange property. Applications to finitely generated modules are also studied. These extend some corresponding results on exchange rings.

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Subprojectivity domains of pure-projective modules

Journal of Algebra and Its Applications ◽

10.1142/s0219498820500917 ◽

2019 ◽

Vol 19 (05) ◽

pp. 2050091

Author(s):

Yılmaz Durğun

Keyword(s):

Noetherian Ring ◽

Projective Module ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Projective Modules ◽

Finitely Generated ◽

Flat Modules ◽

Necessary And Sufficient

In a recent paper, Holston et al. have defined a module [Formula: see text] to be [Formula: see text]-subprojective if for every epimorphism [Formula: see text] and homomorphism [Formula: see text], there exists a homomorphism [Formula: see text] such that [Formula: see text]. Clearly, every module is subprojective relative to any projective module. For a module [Formula: see text], the subprojectivity domain of [Formula: see text] is defined to be the collection of all modules [Formula: see text] such that [Formula: see text] is [Formula: see text]-subprojective. We consider, for every pure-projective module [Formula: see text], the subprojective domain of [Formula: see text]. We show that the flat modules are the only ones sharing the distinction of being in every single subprojectivity domain of pure-projective modules. Pure-projective modules whose subprojectivity domain is as small as possible will be called pure-projective indigent (pp-indigent). Properties of subprojectivity domains of pure-projective modules and of pp-indigent modules are studied. For various classes of modules (such as simple, cyclic, finitely generated and singular), necessary and sufficient conditions for the existence of pp-indigent modules of those types are studied. We characterize the structure of a Noetherian ring over which every (simple, cyclic, finitely generated) pure-projective module is projective or pp-indigent. Furthermore, finitely generated pp-indigent modules on commutative Noetherian hereditary rings are characterized.

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On a Class of Projective Modules Over Central Separable Algebras

Canadian Mathematical Bulletin ◽

10.4153/cmb-1971-072-6 ◽

1971 ◽

Vol 14 (3) ◽

pp. 415-417 ◽

Cited By ~ 2

Author(s):

George Szeto

Keyword(s):

Finite Number ◽

Commutative Ring ◽

Projective Module ◽

Projective Modules ◽

Finitely Generated ◽

Perfect Field ◽

Maximal Ideals ◽

Ring Of Polynomials ◽

Separable Algebras

In [5], DeMeyer extended one consequence of Wedderburn's theorem; that is, if R is a commutative ring with a finite number of maximal ideals (semi-local) and with no idempotents except 0 and 1 or if R is the ring of polynomials in one variable over a perfect field, then there is a unique (up to isomorphism) indecomposable finitely generated projective module over a central separable R-algebra A.

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