The set of singular elements of a ring with respect to a module

Let [Formula: see text] be a ring and [Formula: see text] be a right [Formula: see text]-module. In this paper, we introduce the set [Formula: see text] for some essential submodule [Formula: see text] of [Formula: see text] of singular elements of[Formula: see text] with respect to[Formula: see text] , and we investigate the properties of it. For example, it is shown that [Formula: see text] is an ideal of [Formula: see text] and [Formula: see text]. Also if [Formula: see text] is a semiprime right Goldie ring, then [Formula: see text], where [Formula: see text] is the right singular ideal of [Formula: see text]. We prove that if [Formula: see text] is a semisimple module or a prime module, then [Formula: see text]. For any submodule [Formula: see text] of [Formula: see text], we have [Formula: see text] and if [Formula: see text], then [Formula: see text]. We show that [Formula: see text] and [Formula: see text]. In the end, the singular elements of some rings with respect to the formal triangular matrix ring are investigated.

Twisted Brauer monoids

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.

Presentations for (singular) partition monoids: a new approach

We 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.

On nilpotency of the right singular ideal of semiring

We introduce the concept of nilpotency of the right singular ideal of a semiring. We discuss some properties of such nilpotency and singular ideals. We show that the right singular ideal of a semiring with a.c.c. for right annihilators, is nilpotent.

THE FGF CONJECTURE AND THE SINGULAR IDEAL OF A RING

We show that a left CF ring is left artinian if and only if it is von Neumann regular modulo its left singular ideal. We deduce that a left FGF is Quasi-Frobenius (QF) under this assumption. This clarifies the role played by the Jacobson radical and the singular left ideal in the FGF and CF conjectures. In Sec. 3 of the paper, we study the structure of left artinian left CF rings. We prove that they are left continuous and left CEP rings.

Annihilator-small Right Ideals

A right ideal A of a ring R is called annihilator-small if A+T=R, T a right ideal, implies that [Formula: see text], where [Formula: see text] indicates the left annihilator. The sum Ar of all such right ideals turns out to be a two-sided ideal that contains the Jacobson radical and the left singular ideal, and is contained in the ideal generated by the total of the ring. The ideal Ar is studied, conditions when it is annihilator-small are given, its relationship to the total of the ring is examined, and its connection with related rings is investigated.

SUBSTRUCTURES OF HOM

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.

A Generalization of Semiregular and Almost Principally Injective Rings

In this article, we call a ring R right almost I-semiregular for an ideal I of R if for any a ∈ R, there exists a left R-module decomposition lRrR(a) = P ⊕ Q such that P ⊆ Ra and Q ∩ Ra ⊆ I, where l and r are the left and right annihilators, respectively. This generalizes the right almost principally injective rings defined by Page and Zhou, I-semiregular rings defined by Nicholson and Yousif, and right generalized semiregular rings defined by Xiao and Tong. We prove that R is I-semiregular if and only if for any a ∈ R, there exists a decomposition lRrR(a) = P ⊕ Q, where P = Re ⊆ Ra for some e2 = e ∈ R and Q ∩ Ra ⊆ I. Among the results for right almost I-semiregular rings, we show that if I is the left socle Soc (RR) or the right singular ideal Z(RR) or the ideal Z(RR) ∩ δ(RR), where δ(RR) is the intersection of essential maximal left ideals of R, then R being right almost I-semiregular implies that R is right almost J-semiregular for the Jacobson radical J of R. We show that δl(eRe) = e δ(RR)e for any idempotent e of R satisfying ReR = R and, for such an idempotent, R being right almost δ(RR)-semiregular implies that eRe is right almost δl(eRe)-semiregular.