Two radicals of a ring

Communications in Algebra, 2014

The aim of this paper is to develop an 'external' Kurosh-Amitsur radical theory of semirings and, using this approach, to obtain some fundamental results regarding two Jacobson type of radicals-the Jacobson-Bourne, J-, radical and a very natural its variation, J sradical-of hemirings, as well as the Brown-McCoy, R BM-, radical of hemirings. Among the new central results of the paper, we single out the following ones: Theorems unifying two, internal and external, approches to the Kurosh-Amitzur radical theory of hemirings; A characterization of J-semisimple hemirings; A description of J-semisimple congruence-simple hemirings; A characterization of finite additively-idempotent J s-semisimple hemirings; Complete discriptions of R BM-semisimple commutative and latticeordered hemirings; Semiring versions of the well-known classical ring results-Nakayama's and Hopkins Lemmas and Jacobson-Chevalley Density Theorem; Establishing the fundamental relationship between the radicals J, J s , and R BM of hemirings R and matrix hemirings M n (R); Establishing the matric-extensibleness (see, e.g., [4, Section 4.9]) of the radical classes of the Jacobson, Brown-McCoy, and J s-, radicals of hemirings; Showing that the J-semisimplicity, J s-semisimplicity, and R BM-semisimplicity of semirings are Morita invariant properties.

Acta Mathematica Hungarica, 1996

Throughout, R will denote a commutative ring with identity and every module is unitary. By B < A we mean that B is a proper submodule of A. For the submodules B and C of A, we let (B:C) = {r E R:rC C C_ B}.

Acta Mathematica Hungarica, 2009

Let M be a left R-module. In this paper a generalization of the notion of m-system set of rings to modules is given. Then for a submodule N of M , we dene p √ N := {m ∈ M : every m-system containing m meets N }. It is shown that p √ N is the intersection of all prime submodules of M containing N. We dene radR(M) = p (0). This is called BaerMcCoy radical or prime radical of M. It is shown that if M is an Artinian module over a PI-ring (or an FBN-ring) R, then M/ radR(M) is a Noetherian R-module. Also, if M is a Noetherian module over a PI-ring (or an FBN-ring) R such that every prime submodule of M is virtually maximal, then M/ radR(M) is an Artinian R-module. This yields if M is an Artinian module over a PI-ring R, then either rad R (M) = M or rad R (M) = n i=1 P i M for some maximal ideals P 1 ,. .. , P n of R. Also, Baer's lower nilradical of M [denoted by Nil * (R M)] is dened to be the set of all strongly nilpotent elements of M. It is shown that, for any projective R-module M , radR(M) = Nil * (RM) and, for any module M over a left Artinian ring R, rad R (M) = Nil * (R M) = Rad (M) = Jac (R)M .

Journal of Pure and Applied Algebra, 2015

Given a semigroup S, we prove that if the upper nilradical Nil * (R) is homogeneous whenever R is an S-graded ring, then the semigroup S must be cancelative and torsion-free. In case S is commutative the converse is true. Analogs of these results are established for other radicals and ideals. We also describe a large class of semigroups S with the property that whenever R is a Jacobson radical ring graded by S, then every homogeneous subring of R is also a Jacobson radical ring. These results partially answer two questions of Smoktunowicz. Examples are given delimiting the proof techniques.

Novi Sad Journal of Mathematics, 2018

In this paper we study the notion of regular rings relative to right ideals, and we give another characterization of these rings. Also, we introduce the concept of an annihilator relative to a right ideal. Basic properties of this concept are proved. New results obtained include necessary and sufficient conditions for a ring to be regular (potent) relative to right ideal.

2016

Let R be a ring with identity and J(R) denote the Jacobson radical of R. A ring R is called J-reduced if a = 0 (n ∈ N) implies a ∈ J(R) for any a ∈ R. We give, in this article, many characterizations of such rings. We construct some families of J-reduced rings. Furthermore, the transposes of invertible matrices over such rings are studied.

Notes on the notion of commutative rings as a generalized number systems. Basic constructions are reviewed, including ideals and morphisms. Many "unworked" examples; they turn out to be good exercise for the reader.

Journal of Mathematical Sciences, 2008

Transactions of the American Mathematical Society, 1971

If R is an associative ring, we consider the special Jordan ring R + {R^ + } , and when R has an involution, the special Jordan ring S of symmetric elements. We first show that the prime radical of R equals the prime radical of R + {R^ + } , and that the prime radical of R intersected with S is the prime radical of S. Next we give an elementary characterization, in terms of the associative structure of R, of primeness of S. Finally, we show that a prime ideal of R intersected with S is a prime Jordan ideal of S.

Bulletin of the London Mathematical Society, 2008

A result of Bergman says that the Jacobson radical of a graded algebra is homogeneous. It is shown that while graded Jacobson radical algebras have homogeneous elements nilpotent, it is not the case that graded algebras all of whose homogeneous elements are nilpotent are Jacobson radical. To contrast this, the following result of the author is slightly extended. Let R be a graded algebra generated in the degree one. If for every n, the n × n matrix algebra over R has all homogeneous elements nilpotent, then R is Jacobson radical.