Isomorphisms between endomorphism rings of projective modules

Let R and S be arbitrary rings, RM and SN countably generated free modules, and let φ:End(RM)→End(sN) be an isomorphism between the endomorphism rings of M and N. Camillo [3] showed in 1984 that these assumptions imply that R and S are Morita equivalent rings. Indeed, as Bolla pointed out in [2], in this case the isomorphism φ must be induced by some Morita equivalence between R and S. The same holds true if one assumes that RM and SN are, more generally, non-finitely generated free modules.

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FINITE DOMINATION AND NOVIKOV RINGS. ITERATIVE APPROACH

Glasgow Mathematical Journal ◽

10.1017/s0017089512000419 ◽

2012 ◽

Vol 55 (1) ◽

pp. 145-160 ◽

Cited By ~ 3

Author(s):

THOMAS HÜTTEMANN ◽

DAVID QUINN

Keyword(s):

Laurent Series ◽

Chain Complex ◽

Laurent Polynomial ◽

Projective Modules ◽

Finitely Generated ◽

Formal Laurent Series ◽

Chain Complexes ◽

Laurent Polynomial Ring ◽

Bounded Below ◽

Free Modules

AbstractSuppose C is a bounded chain complex of finitely generated free modules over the Laurent polynomial ring L = R[x,x−1]. Then C is R-finitely dominated, i.e. homotopy equivalent over R to a bounded chain complex of finitely generated projective R-modules if and only if the two chain complexes C ⊗LR((x)) and C ⊗LR((x−1)) are acyclic, as has been proved by Ranicki (A. Ranicki, Finite domination and Novikov rings, Topology34(3) (1995), 619–632). Here R((x)) = R[[x]][x−1] and R((x−1)) = R[[x−1]][x] are rings of the formal Laurent series, also known as Novikov rings. In this paper, we prove a generalisation of this criterion which allows us to detect finite domination of bounded below chain complexes of projective modules over Laurent rings in several indeterminates.

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Endomorphism rings of finitely generated projective modules

Pacific Journal of Mathematics ◽

10.2140/pjm.1973.47.199 ◽

1973 ◽

Vol 47 (1) ◽

pp. 199-220 ◽

Cited By ~ 3

Author(s):

Robert Miller

Keyword(s):

Projective Modules ◽

Finitely Generated ◽

Endomorphism Rings

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Endomorphism Rings and Gabriel Topologies

Canadian Journal of Mathematics ◽

10.4153/cjm-1984-013-4 ◽

1984 ◽

Vol 36 (2) ◽

pp. 193-205 ◽

Cited By ~ 1

Author(s):

Soumaya Makdissi Khuri

Keyword(s):

Endomorphism Ring ◽

Lattice Isomorphism ◽

Close Connection ◽

Projective Modules ◽

Finitely Generated ◽

Basic Tool ◽

Endomorphism Rings ◽

Finite Dimensional ◽

Faithful Module ◽

Correspondence Theorem

A basic tool in the usual presentation of the Morita theorems is the correspondence theorem for projective modules. Let RM be a left R-module and B = HomR(M, M). When M is a progenerator, there is a close connection (in fact a lattice isomorphism) between left R-submodules of M and left ideals of B, which can be applied to the solution of problems such as characterizing when the endomorphism ring of a finitely generated projective faithful module is simple or right Noetherian. More generally, Faith proved that this connection can be retained in suitably modified form when M is just a generator in R-mod ([4], [2], [3]). In this form the correspondence theorem can be applied to show, e.g., that, when RM is a generator, then (a): RM is finite-dimensional if and only if B is a left finite-dimensional ring and in this case d(RM) = d(BB), and (b): If RM is nonsingular then B is a left nonsingular ring ([6]).

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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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