Direct products of modules whose endomorphism rings have at most two maximal ideals

AbstractLet R be a ring and U a left R-module with S=End(RU). The aim of this paper is to characterize when S is coherent. We first show that a left R-module F is TU-flat if and only if HomR(U,F) is a flat left S-module. This removes the unnecessary hypothesis that U is Σ-quasiprojective from Proposition 2.7 of Gomez Pardo and Hernandez [‘Coherence of endomorphism rings’, Arch. Math. (Basel)48(1) (1987), 40–52]. Then it is shown that S is a right coherent ring if and only if all direct products of TU-flat left R-modules are TU-flat if and only if all direct products of copies of RU are TU-flat. Finally, we prove that every left R-module is TU-flat if and only if S is right coherent with wD(S)≤2 and US is FP-injective.

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Central lifting property for orthomodular lattices

Mathematica Slovaca ◽

10.1515/ms-2017-0433 ◽

2020 ◽

Vol 70 (6) ◽

pp. 1307-1316

Author(s):

Neda Arjomand Kermani ◽

Esfandiar Eslami ◽

Arsham Borumand Saeid

Keyword(s):

Boolean Algebras ◽

Prime Ideals ◽

Direct Products ◽

Orthomodular Lattices ◽

Maximal Ideals ◽

Central Elements ◽

Simple Chain

AbstractWe introduce and investigate central lifting property (CLP) for orthomodular lattices as a property whereby all central elements can be lifted modulo every p-ideal. It is shown that prime ideals, maximal ideals and finite p-ideals have CLP. Also Boolean algebras, simple chain finite orthomodular lattices, subalgebras of an orthomodular lattices generated by two elements and finite orthomodular lattices have CLP. The main results of the present paper include the investigation of CLP for principal p-ideals and finite direct products of orthomodular lattices.

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KERNELS OF MORPHISMS BETWEEN INDECOMPOSABLE INJECTIVE MODULES

Glasgow Mathematical Journal ◽

10.1017/s0017089510000170 ◽

2010 ◽

Vol 52 (A) ◽

pp. 69-82 ◽

Cited By ~ 13

Author(s):

ALBERTO FACCHINI ◽

ŞULE ECEVIT ◽

M. TAMER KOŞAN

Keyword(s):

Local Ring ◽

Endomorphism Ring ◽

Weak Form ◽

Indecomposable Module ◽

Endomorphism Rings ◽

Direct Sums ◽

Indecomposable Modules ◽

Direct Sum ◽

Injective Modules ◽

Maximal Ideals

AbstractWe show that the endomorphism rings of kernels ker ϕ of non-injective morphisms ϕ between indecomposable injective modules are either local or have two maximal ideals, the module ker ϕ is determined up to isomorphism by two invariants called monogeny class and upper part, and a weak form of the Krull–Schmidt theorem holds for direct sums of these kernels. We prove with an example that our pathological decompositions actually take place. We show that a direct sum ofnkernels of morphisms between injective indecomposable modules can have exactlyn! pairwise non-isomorphic direct-sum decompositions into kernels of morphisms of the same type. IfERis an injective indecomposable module andSis its endomorphism ring, the duality Hom(−,ER) transforms kernels of morphismsER→ERinto cyclically presented left modules over the local ringS, sending the monogeny class into the epigeny class and the upper part into the lower part.

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IDEALS IN DIRECT PRODUCTS OF COMMUTATIVE RINGS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972708000415 ◽

2008 ◽

Vol 77 (3) ◽

pp. 477-483

Author(s):

D. D. ANDERSON ◽

JOHN KINTZINGER

Keyword(s):

Abelian Group ◽

Maximal Ideal ◽

Commutative Rings ◽

Primary Ideal ◽

Maximal Subgroups ◽

Prime Ideals ◽

Direct Products ◽

Maximal Ideals ◽

Radical Ideals ◽

Primary Ideals

AbstractLet R and S be commutative rings, not necessarily with identity. We investigate the ideals, prime ideals, radical ideals, primary ideals, and maximal ideals of R×S. Unlike the case where R and S have an identity, an ideal (or primary ideal, or maximal ideal) of R×S need not be a ‘subproduct’ I×J of ideals. We show that for a ring R, for each commutative ring S every ideal (or primary ideal, or maximal ideal) is a subproduct if and only if R is an e-ring (that is, for r∈R, there exists er∈R with err=r) (or u-ring (that is, for each proper ideal A of R, $\sqrt {A}\not =R$)), the Abelian group (R/R2 ,+) has no maximal subgroups).

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Finitary ideals of direct products in quantales

Revista de Ciência da Computação ◽

10.22481/recic.v3i1.8941 ◽

2021 ◽

Vol 3 (1) ◽

pp. 23-28

Author(s):

Pascal Pankiti ◽

C Nkuimi-Jugnia

Keyword(s):

Complete Lattice ◽

Tensor Category ◽

Prime Ideals ◽

Direct Products ◽

C Algebra ◽

Maximal Ideals ◽

The Common ◽

Radical Ideals ◽

Primary Ideals ◽

Unital Rings

The notion of quantale, which designates a complete lattice equipped with an associative binary multiplication distributing over arbitrary joins, appears in various areasof mathematics-in quantaloid theory, in non classical logic as completion of the Lindebaum algebra, and in different representations of the spectrum of a C∗ algebra asmany-valued and non commutative topologies. To put it briefly, its importance is nolonger to be demonstrated. Quantales are ring-like structures in that they share withrings the common fact that while as rings are semi groups in the tensor category ofabelian groups, so quantales are semi groups in the tensor category of sup-lattices.In 2008 Anderson and Kintzinger [1] investigated the ideals, prime ideals, radical ideals, primary ideals, and maximal of a product ring R × S of two commutativenon non necceray unital rings R and S: Something resembling rings are quantales byanalogy with what is studied in ring, we begin an investigation on ideals of a productof two quantales. In this paper, given two quantales Q1 and Q2; not necessarily withidentity, we investigate the ideals, prime ideals, primary ideals, and maximal ideals ofthe quantale Q1 × Q2

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