scholarly journals Every Semiprimary Ring is the Endomorphism Ring of a Projective Module Over a Quasi-Hereditary Ring

1989 ◽  
Vol 107 (1) ◽  
pp. 1 ◽  
Author(s):  
Vlastimil Dlab ◽  
Claus Michael Ringel
Keyword(s):  
Endomorphism Ring ◽  
Projective Module ◽  
Hereditary Ring ◽  
Semiprimary Ring

scholarly journals On Frobenius Extensions I.

1960 ◽  
Vol 17 ◽  
pp. 89-110 ◽  
Author(s):  
Tadasi Nakayama ◽  
Tosiro Tsuzuku
Keyword(s):  
Commutative Ring ◽  
Endomorphism Ring ◽  
Projective Module ◽  
Commutative Rings ◽  
Free Module ◽  
General Tendency ◽  
General Notion ◽  
Frobenius Algebras ◽  
Definition Of

As a generalization of the notion of Frobenius algebras over a field Kasch [103 introduced that of Frobenius extensions of a ring. The present writers [13] recently freed one of Kasch’s main theorems from its rather strong S-ring assumption of the ground ring. However, even with the removal of the S-ring assumption of the ground ring the notion does not seem general enough, and we wish, in the present paper and its sequel, to develope the theory upon the basis of a more general notion of Frobenius extensions. Thus, we replace the free module property of the extension by the projective module property (according to a general tendency in algebra), which has been done in fact in case of Frobenius algebras over a commutative ring in a previous work by Eilenberg and one of the writers [4], and, further, take automorphisms of the ground ring into the definition of Frobenius extensions (which seems quite natural particularly in case of non-commutative rings). To such generalized notion of Frobenius extensions we may extend many of Kasch’s theorems, including those which are immediate extensions of classical theorems for Frobenius algebras and those which are essentially new, as the above alluded endomorphism ring theorem. Also homological properties of Frobenius extensions, as were developed in Hirata’s [6] recent paper in succession to Eilenberg-Nakayama [4], can be extended to our present generalized case; we shall also exceed [4], [6] somewhat in considering injective and weak dimensions.

Σ-Rickart modules

2019 ◽  
Vol 19 (11) ◽  
pp. 2050207
Author(s):  
Gangyong Lee ◽  
Mauricio Medina-Bárcenas
Keyword(s):  
Direct Summand ◽  
Endomorphism Ring ◽  
Finitely Generated ◽  
Hereditary Ring ◽  
Direct Sum ◽  
Factor Module ◽  
Rickart Modules

Hereditary rings have been extensively investigated in the literature after Kaplansky introduced them in the earliest 50’s. In this paper, we study the notion of a [Formula: see text]-Rickart module by utilizing the endomorphism ring of a module and using the recent notion of a Rickart module, as a module theoretic analogue of a right hereditary ring. A module [Formula: see text] is called [Formula: see text]-Rickart if every direct sum of copies of [Formula: see text] is Rickart. It is shown that any direct summand and any direct sum of copies of a [Formula: see text]-Rickart module are [Formula: see text]-Rickart modules. We also provide generalizations in a module theoretic setting of the most common results of hereditary rings: a ring [Formula: see text] is right hereditary if and only if every submodule of any projective right [Formula: see text]-module is projective if and only if every factor module of any injective right [Formula: see text]-module is injective. Also, we have a characterization of a finitely generated [Formula: see text]-Rickart module in terms of its endomorphism ring. Examples which delineate the concepts and results are provided.

ON M-C-INJECTIVE AND SELF-C-INJECTIVE MODULES

2010 ◽  
Vol 03 (03) ◽  
pp. 387-393 ◽  
Author(s):  
A. K. Chaturvedi ◽  
B. M. Pandeya ◽  
A. M. Tripathi ◽  
O. P. Mishra
Keyword(s):  
Endomorphism Ring ◽  
Projective Module ◽  
Injective Module ◽  
Injective Modules ◽  
Factor Module ◽  
Closed Submodule

Let M1 and M2 be two R-modules. Then M2 is called M1-c-injective if every homomorphism α from K to M2, where K is a closed submodule of M1, can be extended to a homomorphism β from M1 to M2. An R-module M is called self-c-injective if M is M-c-injective. For a projective module M, it has been proved that the factor module of an M -c-injective module is M -c-injective if and only if every closed submodule of M is projective. A characterization of self-c-injective modules in terms endomorphism ring of an R-module satisfying the CM-property is given.

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

1988 ◽  
Vol 30 (2) ◽  
pp. 215-220 ◽  
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].

scholarly journals Coranks of a quasi-projective module and its endomorphism ring

1994 ◽  
Vol 36 (3) ◽  
pp. 381-383 ◽  
Author(s):  
Tsutomu Takeuchi
Keyword(s):  
Endomorphism Ring ◽  
Projective Module ◽  
Present Note ◽  
Goldie Dimension

Recently several authors have studied dualizing Goldie dimension of a module: spanning dimension in [2], codimension in [13], corank in [16] and also [9,17,12, 5,11, 6, 4, 7] ([13] may be read in comparison with the others). In the present note we prove the equality corank RP = corank SS, where P is a quasi-projective left R-module and S is its endomorphism ring. This result is an answer to the question [12, p. 1898] and an extension of [3, Corollary 4.3] which shows the above equality for a Σ-quasi-projective left R-module P.

scholarly journals THE UNIT SUM NUMBER OF SOME PROJECTIVE MODULES

2008 ◽  
Vol 50 (1) ◽  
pp. 71-74
Author(s):  
NAHID ASHRAFI
Keyword(s):  
Endomorphism Ring ◽  
Projective Module ◽  
Projective Modules ◽  
Perfect Ring ◽  
Unit Sum Number

AbstractThe unit sum number u(R) of a ring R is the least k such that every element is the sum of k units; if there is no such k then u(R) is ω or ∞ depending whether the units generate R additively or not. If RM is a left R-module, then the unit sum number of M is defined to be the unit sum number of the endomorphism ring of M. Here we show that if R is a ring such that R/J(R) is semisimple and $\Z_{2}$ is not a factor of R/J(R) and if P is a projective R-module such that JP ≪ P, (JP small in P), then u(P)= 2. As a result we can see that if P is a projective module over a perfect ring then u(P)=2.

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