scholarly journals The singular ideal and the socle of incidence rings

2020 ◽  
pp. 1-18
Author(s):  
Müge KANUNİ ER ◽  
Özkay ÖZKAN
Keyword(s):  
Singular Ideal

scholarly journals Presentations for (singular) partition monoids: a new approach

2017 ◽  
Vol 165 (3) ◽  
pp. 549-562 ◽  
Author(s):  
JAMES EAST
Keyword(s):  
New Approach ◽  
Singular Ideal

AbstractWe give new, short proofs of the presentations for the partition monoid and its singular ideal originally given in the author's 2011 papers inJournal of AlgebraandInternational Journal of Algebra and Computation.

The Singular Congruence and the Maximal Quotient Semigroup

1972 ◽  
Vol 15 (2) ◽  
pp. 301-303 ◽  
Author(s):  
F. R. McMorris
Keyword(s):  
Quotient Ring ◽  
Von Neumann ◽  
Von Neumann Regular ◽  
Singular Ideal ◽  
Quotient Semigroup

It is a well known result (see [4, p. 108]) that if R is a ring and Q(R) its maximal right quotient ring, then Q(R) is (von Neumann) regular if and only if every large right ideal of R is dense. This condition is equivalent to saying that the singular ideal of R is zero. In this note we show that the condition loses its magic in the theory of semigroups.

Note on the Singular Submodule

1970 ◽  
Vol 13 (4) ◽  
pp. 441-442
Author(s):  
D. Fieldhouse
Keyword(s):  
Ring Theory ◽  
New Technique ◽  
Ring Homomorphisms ◽  
A New Technique ◽  
Singular Ideal

One very interesting and important problem in ring theory is the determination of the position of the singular ideal of a ring with respect to the various radicals (Jacobson, prime, Wedderburn, etc.) of the ring. A summary of the known results can be found in Faith [3, p. 47 ff.] and Lambek [5, p. 102 ff.]. Here we use a new technique to obtain extensions of these results as well as some new ones.Throughout we adopt the Bourbaki [2] conventions for rings and modules: all rings have 1, all modules are unital, and all ring homomorphisms preserve the 1.

Regular, Commutative, Maximal Semigroups of Quotients

1975 ◽  
Vol 18 (1) ◽  
pp. 99-104 ◽  
Author(s):  
Jurgen Rompke
Keyword(s):  
Commutative Semigroup ◽  
Regular Ring ◽  
Sufficient Conditions ◽  
Commutative Rings ◽  
Necessary And Sufficient Conditions ◽  
Natural Question ◽  
Commutative Case ◽  
Ring Of Quotients ◽  
Singular Ideal ◽  
Necessary And Sufficient

A well-known theorem which goes back to R. E. Johnson [4], asserts that if R is a ring then Q(R), its maximal ring of quotients is regular (in the sense of v. Neumann) if and only if the singular ideal of R vanishes. In the theory of semigroups a natural question is therefore the following: Do there exist properties which characterize those semigroups whose maximal semigroups of quotients are regular? Partial answers to this question have been given in [3], [7] and [8]. In this paper we completely solve the commutative case, i.e. we give necessary and sufficient conditions for a commutative semigroup S in order that Q(S), the maximal semigroup of quotients, is regular. These conditions reflect very closely the property of being semiprime, which in the theory of commutative rings characterizes those rings which have a regular ring of quotients.

scholarly journals SUBSTRUCTURES OF HOM

2011 ◽  
Vol 10 (01) ◽  
pp. 119-127 ◽  
Author(s):  
TSIU-KWEN LEE ◽  
YIQIANG ZHOU
Keyword(s):  
Jacobson Radical ◽  
The Other ◽  
Natural Question ◽  
Singular Ideal

Let M and N be two modules over a ring R. The concern is about the four substructures of hom R(M, N): the Jacobson radical J[M, N], the singular ideal Δ[M, N], the co-singular ideal ∇[M, N] and the total Tot [M, N]. One natural question is to characterize when the total is equal to one or more of the other structures. We review some known results and prove several new results towards this question and, as consequences, give answers to some related questions.

Pi-Rings with Zero Singular Ideal

1973 ◽  
Vol s2-6 (4) ◽  
pp. 629-632 ◽  
Author(s):  
A. W. Chatters
Keyword(s):  
Singular Ideal

On Modules of Singular Submodule Zero

1971 ◽  
Vol 23 (2) ◽  
pp. 345-354 ◽  
Author(s):  
Vasily C. Cateforis ◽  
Francis L. Sandomierski
Keyword(s):  
Integral Domain ◽  
Associative Ring ◽  
Quotient Ring ◽  
Essential Extension ◽  
Singular Ideal ◽  
Free Modules ◽  
Torsion Free ◽  
Unitary Right

In this paper we generalize to modules of singular submodule zero over a ring of singular ideal zero some of the results, which are well known for torsion-free modules over a commutative integral domain, e.g. [2, Chapter VII, p. 127], or over a ring, which possesses a classical right quotient ring, e.g. [13, § 5].Let R be an associative ring with 1 and let M be a unitary right R-module, the latter fact denoted by MR. A submodule NR of MR is large in MR (MR is an essential extension of NR) if NR intersects non-trivially every non-zero submodule of MR; the notation NR ⊆′ MR is used for the statement “NR is large in MR” The singular submodule of MR, denoted Z(MR), is then defined to be the set {m ∈ M| r(m) ⊆’ RR}, where

On Ring Properties of Injective Hulls

1975 ◽  
Vol 18 (2) ◽  
pp. 233-239 ◽  
Author(s):  
N. C. Lang
Keyword(s):  
Regular Ring ◽  
Associative Ring ◽  
Injective Hull ◽  
Von Neumann ◽  
Singular Ideal ◽  
The Right ◽  
Injective Hulls

Let R be an associative ring and denote by the injective hull of the right module RR. If can be endowed with a ring multiplication which extends the existing module multiplication, we say that is a ring and the statement that R is a ring will always mean in this sense.It is known that is a regular ring (in the sense of von Neumann) if and only if the singular ideal of R is zero.

Twisted Brauer monoids

2018 ◽  
Vol 148 (4) ◽  
pp. 731-750 ◽  
Author(s):  
Igor Dolinka ◽  
James East
Keyword(s):  
Principal Ideal ◽  
Sufficient Conditions ◽  
Singular Part ◽  
Necessary And Sufficient Conditions ◽  
Green’S Relations ◽  
Generating Set ◽  
Idempotent Rank ◽  
Singular Ideal ◽  
Necessary And Sufficient ◽  
Lattice Of Ideals

We investigate the structure of the twisted Brauer monoid , comparing and contrasting it with the structure of the (untwisted) Brauer monoid . We characterize Green's relations and pre-orders on , describe the lattice of ideals and give necessary and sufficient conditions for an ideal to be idempotent generated. We obtain formulae for the rank (smallest size of a generating set) and (where applicable) the idempotent rank (smallest size of an idempotent generating set) of each principal ideal; in particular, when an ideal is idempotent generated, its rank and idempotent rank are equal. As an application of our results, we describe the idempotent generated subsemigroup of (which is not an ideal), as well as the singular ideal of (which is neither principal nor idempotent generated), and we deduce that the singular part of the Brauer monoid is idempotent generated, a result previously proved by Maltcev and Mazorchuk.

scholarly journals Weak $q$-rings with zero singular ideal

1979 ◽  
Vol 76 (1) ◽  
pp. 25-25
Author(s):  
Saad Mohamed ◽  
Surjeet Singh
Keyword(s):  
Singular Ideal
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