Generalized Matrix Ring Research Articles

Partial actions of groups on generalized matrix rings

Let n be a positive integer and R=(Mij)1≤i,j≤n be a generalized matrix ring. For each 1≤i,j≤n, let Ii be an ideal of the ring Ri:=Mii and denote Iij=IiMij+MijIj. We give sufficient conditions for the subset I=(Iij)1≤i,j≤n of R to be an ideal of R. Also, suppose that α(i) is a partial action of a group G on Ri, for all 1≤i≤n. We construct, under certain conditions, a partial action γ of G on R such that γ restricted to Ri coincides with α(i). We study the relation between this construction and the notion of Morita equivalent partial group action given in [1]. Moreover, we investigate properties related to Galois theory for the extension Rγ⊂R. Some examples to illustrate the results are considered in the last part of the paper.

FI-extending generalized matrix rings

Recall that a module [Formula: see text] is FI-extending if every fully invariant submodule is essential in a direct summand of [Formula: see text]. Let [Formula: see text] be a generalized matrix ring, where [Formula: see text] and [Formula: see text] are rings and [Formula: see text] and [Formula: see text] are bimodules. In this paper, we investigate necessary and/or sufficient conditions on [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] for [Formula: see text] to be FI-extending.

Structure of Zhou Nil-clean Rings

A ring R is Zhou nil-clean if every element in R is the sum of two tripotents and a nilpotent that commute. Homomorphic images of Zhou nil-clean rings are explored. We prove that a ring R is Zhou nil-clean if and only if 30 ϵ R is nilpotent and R/30R is Zhou nil-clean, if and only if R/BM(R) is 5-potent and BM(R) is nil, if and only if J(R) is nil and R/J(R) is isomorphic to a Boolean ring, a Yaqub ring, a Bell ring or a direct product of such rings. By means of homomorphic images, we completely determine when the generalized matrix ring is Zhou nil-clean. We prove that the generalized matrix ring Mn(R; s) is Zhou nil-clean if and only if R is Zhou nil-clean and s ϵ J(R).

Characterizations of Jordan mappings on some rings and algebras through zero products

Let be a generalized matrix ring, where and are 2-torsion free. We prove that if is an additive mapping such that whenever then , where is a Jordan derivation and is a multiplier. As its applications, we prove that the similar conclusion remains valid on full matrix algebras, unital prime rings with a nontrivial idempotent, unital standard operator algebras, CDCSL algebras and von Neumann algebras.

Idempotents and structures of rings

Recall that an n-by-n generalized matrix ring is defined in terms of sets of rings -bimodules and bimodule homomorphisms , where the set of diagonal matrix units form a complete set of orthogonal idempotents. Moreover, an arbitrary ring with a complete set of orthogonal idempotents has a Peirce decomposition which can be arranged into an n-by-n generalized matrix ring which is isomorphic to R. In this paper, we focus on the subclass of n-by-n generalized matrix rings with for . contains all upper and all lower generalized triangular matrix rings. The triviality of the bimodule homomorphisms motivates the introduction of three new types of idempotents called the inner Peirce, outer Peirce and Peirce trivial idempotents. These idempotents are our main tools and are used to characterize and define a new class of rings called the n-Peirce rings. If R is an n-Peirce ring, then there is a certain complete set of orthogonal idempotents such that . We show that every n-by-n generalized matrix ring R contains a subring S which is maximal with respect to being in and S is essential in R as an (S, S)-bisubmodule of R. This allows for a useful transfer of information between R and S. Also, we show that any ring is either an n-Peirce ring or for each there is a complete set of orthogonal idempotents such that . Examples are provided to illustrate and delimit our results.

Calculation of the group K 1 of a generalized matrix ring

We express the Whitehead group K1 of an arbitrary generalized matrix ring in terms of the K1 group of the original rings.

Quasipolar Property of Generalized Matrix Rings

The article concerns the question of when a generalized matrix ring K s (R) over a local ring R is quasipolar. For a commutative local ring R, it is proved that K s (R) is quasipolar if and only if it is strongly clean. For a general local ring R, some partial answers to the question are obtained. There exist noncommutative local rings R such that K s (R) is strongly clean, but not quasipolar. Necessary and sufficient conditions for a single matrix of K s (R) (where R is a commutative local ring) to be quasipolar is obtained. The known results on this subject in [5] are improved or extended.

Study of Morita contexts

This article concerns mainly on various ring properties of Morita contexts. Necessary and sufficient conditions are obtained for a general Morita context or a trivial Morita context or a generalized matrix ring over a ring to satisfy a certain ring property which is among being semilocal, semiperfect, left perfect, semiprimary, semipotent, potent, clean, strongly π-regular, semiregular, etc. Many known results on a formal triangular matrix ring are extended to a Morita context or a trivial Morita context. Some questions on this subject raised by Varadarajan in [22] are answered.

The group K 0 of a generalized matrix ring

We deal with a group K 0 of some category of modules over a generalized matrix ring (of order 2). The results obtained are applied to compute the group K 0 for the generalized matrix ring itself.

Strong cleanness of generalized matrix rings over a local ring

Let Ks(R) be the generalized matrix ring over a ring R with multiplier s. For a general local ring R and a central element s in the Jacobson radical of R, necessary and sufficient conditions are obtained for Ks(R) to be a strongly clean ring. For a commutative local ring R and an arbitrary element s in R, criteria are obtained for a single element of Ks(R) to be strongly clean and, respectively, for the ring Ks(R) to be strongly clean. Specializing to s=1 yields some known results. New families of strongly clean rings are presented.

Ideals of generalized matrix rings

Let and be rings, and and bimodules. In the paper, in terms of isomorphisms of lattices, relationships between the lattices of one-sided and two-sided ideals of the generalized matrix ring and the corresponding lattices of ideals of the rings and are described. Necessary and sufficient conditions for a pair of ideals , of rings and , respectively, to be the main diagonal of some ideal of the ring are also obtained.Bibliography: 8 titles.

Left Self Distributively Generated Structural Matrix Rings#

ABSTRACT Let K denote a commutative ring with unity and A be a K-algebra. An element, d ∈ A is said to be left self distributive, or LSD, if dxy = dx dy for all x, y ∈ A. Let ℒ(A) be the set of LSD elements. Similarly, one can define the set of right self distributive, or RSD, elements and let ℛ(A) be the set of RSD elements. Let 𝒟(A) = ℒ(A) ∩ ℛ(A), the set of self distributive, or SD, elements. An algebra, A, is said to be left self distributively generated, or LSD-generated, if A = mod K (ℒ(A)), the K-module generated by ℒ(A). Analogously, one defines RSD-generated and SD-generated algebras. If A = mod K (ℒ(A)) = mod K (ℛ(A)), then A is said to be LSD/RSD-generated, which is a strictly larger class than the class of SD-generated algebras. Examples are given to illustrate the variety of LSD-generated algebras. This paper continues the study of LSD-generated, RSD-generated, LSD/RSD-generated and SD-generated algebras. This paper characterizes exactly which structural matrix rings are LSD-generated. The paper begins with an important lemma that characterizes LSD elements in a matrix ring in terms of the entries of the matrix. The main result characterizes those structural matrix rings that are LSD-generated, first in terms of a 2 × 2 generalized matrix ring, then strictly in terms of the shape of the matrix ring. Sharper results are obtained for LSD/RSD-generated and SD-generated structural matrix rings. The final section is devoted to an application of this result to endomorphism rings. If the endomorphism ring of a finitely generated module is a homomorphic image of a structural matrix ring, then the module is a direct sum of cyclic modules. Further conditions are given to describe when the structural matrix ring is LSD-generated, in terms of the annihilators of the generating set.

Net subrings of generalized matrix rings

Let Λ be a ring with the following properties: (a) Λ is a direct sum of left ideals P1,..., Pn; (b) every nontrivial homomorphism Pi→Pj is a monomorphism; (c) for every i, j the intersection of any two submodules of Pj isomorphic to Pi contains a submodule isomorphic to Pi. Then Λ can be represented as a subring associated with a net of ideals in a generalized matrix ring. Bibliography: 5 titles.

Classical quotient rings of generalized matrix rings

An associative ring R with identity is a generalized matrix ring with idempotent set E if E is a finite set of orthogonal idempotents of R whose sum is 1. We show that, in the presence of certain annihilator conditions, such a ring is semiprime right Goldie if and only if eRe is semiprime right Goldie for all e ∈ E, and we calculate the classical right quotient ring of R.