Formal Matrix Rings Research Articles

Involutions of <i>K<sub>s</sub>(R)</i>

We give a description of involutions in formal matrix ring Ks(R) over an UFD. Some results on equivalency of involutions were also obtained.

Formal matrix rings: Automorphisms

Automorphism groups of formal matrix algebras with zero trace ideals are studied. Such an algebra is represented as a splitting extension of some ring by some nilpotent ideal. Using this extension, the study of the structure of the automorphism group of an algebra in a certain sense is reduced to the study of the structure of some its subgroups and quotient groups. Then the structure of these subgroups and factor groups is found. The case of formal triangular matrix rings is specially considered. At the end of the paper we study automorphisms of ordinary matrix rings. In this case, a well-known homomorphism from the group of outer automorphisms to the Picard group of a certain ring is used.

Formal Matrix Rings: Isomorphism Problem

We consider the isomorphism problem for formal matrix rings over a given ring. Principal factor matrices of such rings play an important role in this case.

On the automorphism-invariance of finitely generated ideals and formal matrix rings

In this paper, we study rings having the property that every finitely generated right ideal is automorphism-invariant. Such rings are called right f a-rings. It is shown that a right f a-ring with finite Goldie dimension is a direct sum of a semisimple artinian ring and a basic semiperfect ring. Assume that R is a right f a-ring with finite Goldie dimension such that every minimal right ideal is a right annihilator, its right socle is essential in RR, R is also indecomposable (as a ring), not simple, and R has no trivial idempotents. Then R is QF. In this case, QF-rings are the same as q?, f q?, a?, f a-rings. We also obtain that a right module (X,Y, f, g) over a formal matrix ring (R M N S) with canonical isomorphisms f? and g? is automorphism-invariant if and only if X is an automorphism-invariant right R-module and Y is an automorphism-invariant right S-module.

On realization and isomorphism problems for formal matrix rings

On realization and isomorphism problems for formal matrix rings

On UQ rings

In this paper, we introduce [Formula: see text] rings. A ring [Formula: see text] is called [Formula: see text] if every non-unit element of [Formula: see text] can be represented as a product of a unit and a quasiregular element. We provide various properties of [Formula: see text] rings along with its characterizations. We give a new characterization of [Formula: see text]-good rings, and it turns out that [Formula: see text]-good rings are precisely the rings in which every element is a product of a unit and a quasiregular element. We discuss various extensions of [Formula: see text] rings such as Morita contexts, generalized matrix rings, formal matrix rings, group rings, etc.

On Some Class of Formal Matrix Ring

On Some Class of Formal Matrix Ring

О fc-ниль-хороших кольцах формальных матриц

In 2018, Abdolyusefi, Ashrafi, and Chen gave a definition of a 2-nil-good ring element in their work, generalizing the notion of a graceful ring element introduced two years earlier by Kalugeryan and Lam, as well as the definition of a 2-nil-good ring. In the same work, it was shown that the Morita context ring, i.e. a formal matrix ring of the second order is 2-nil-good if the rings over which it is considered are themselves 2-nil-good. In this paper, we generalize further, defining k-nil-good elements and k-nil-good rings, and state a condition under which a formal matrix ring of an arbitrary finite order is k-nil-good.

K-Good Formal Matrix Rings of Infinite Order

k-Good Formal Matrix Rings of Infinite Order

Rings all of whose finitely generated ideals are automorphism-invariant

Rings in which each finitely generated right ideal is automorphism-invariant (right[Formula: see text]-rings) are shown to be isomorphic to a formal matrix ring. Among other results it is also shown that (i) if [Formula: see text] is a right nonsingular ring and [Formula: see text] is an integer, then [Formula: see text] is a right self injective regular ring if and only if the matrix ring [Formula: see text] is a right [Formula: see text]-ring, if and only if [Formula: see text] is a right automorphism-invariant ring and (ii) a right nonsingular ring [Formula: see text] is a right [Formula: see text]-ring if and only if [Formula: see text] is a direct sum of a square-full von Neumann regular right self-injective ring and a strongly regular ring containing all invertible elements of its right maximal ring of fractions. In particular, we show that a right semiartinian (or left semiartinian) ring [Formula: see text] is a right nonsingular right [Formula: see text]-ring if and only if [Formula: see text] is a left nonsingular left [Formula: see text]-ring.

Simple-Direct Modules Over Formal Matrix Rings

In the present paper, we study simple-direct-injective modules and simple-direct-projective modules over a formal matrix ring $$K=\left(\begin{matrix}R&M\\ N&S\end{matrix}\right)$$ , where $$M$$ is an $$(R,S)$$ -bimodule and $$N$$ is a $$(S,R)$$ -bimodule over rings $$R$$ and $$S$$ . We determine necessary and sufficient conditions for a $$K$$ -module to be, respectively, simple-direct-injective or simple-direct-projective. We also give some examples to illustrate and delimit our results.

English

We investigate the formal matrix ring over $R$ defined by a certain system of factors. We give a method for constructing formal matrix rings from non-negative integer matrices. We also show that the principal factor matrix of a binary system of factors determine the structure of the system.

Formal Matrices and Rings Close to Regular

This paper contains new and known results on formal matrix rings close to regular. The main results are given with proofs.

Isomorphisms of Formal Matrix Rings with Zero Trace Ideals

We obtain explicit criteria for the isomorphism of formal matrix rings with zero trace ideals. In particular, we consider the case of formal upper-triangular matrix rings with semicentral reduced rings on the principal diagonal.

Isomorphisms of formal matrix incidence rings

Abstract—In a paper published in 2008 P. A. Krylov showed that formal matrix rings Ks(R) and Kt(R) are isomorphic if and only if the elements s and t differ up to an automorphism by an invertible element. Similar dependence takes place in many cases. In this paper we consider formal matrix rings (and algebras) which have the same structure as incidence rings. We show that the isomorphism problem for formal matrix incidence rings can be reduced to the isomorphism problem for generalized incidence algebras. For these algebras, the direct assertion of Krylov’s theorem holds, but the converse is not true. In particular, we obtain a complete classification of isomorphisms of generalized incidence algebras of order 4 over a field. We also consider the isomorphism problem for special classes of formal matrix rings, namely, formal matrix rings with zero trace ideals.

Grothendieck and Whitehead Groups of Formal Matrix Rings

Grothendieck and Whitehead Groups of Formal Matrix Rings

SSP rings and modules

We study [Formula: see text]-Ci modules ([Formula: see text]), first introduced in [K. Oshiro, Continuous modules and quasi-continuous modules, Osaka J. Math. 20 (1983) 681–694], and [Formula: see text]-SSP modules. We consider the cases when these classes of modules coincide. As a consequence, we obtain some results related to simple-direct-injective modules. We also investigate some properties of SSP formal matrix rings and describe semiartinian SSP rings.

Formal matrix rings and their isomorphisms

We study the isomorphism problem for formal matrix rings and obtain the description of semiartinian formal matrix rings and the max-rings of formal matrices.

Formal Matrices and Their Determinants

In the present paper, we study formal matrix rings over a given ring and determinants of such matrices.

Nil-clean and strongly nil-clean rings

An element a of a ring R is nil-clean if a=e+b where e2=e∈R and b is a nilpotent; if further eb=be, the element a is called strongly nil-clean. The ring R is called nil-clean (resp., strongly nil-clean) if each of its elements is nil-clean (resp., strongly nil-clean). It is proved that an element a is strongly nil-clean iff a is a sum of an idempotent and a unit that commute and a−a2 is a nilpotent, and that a ring R is strongly nil-clean iff R/J(R) is boolean and J(R) is nil, where J(R) denotes the Jacobson radical of R. The strong nil-cleanness of Morita contexts, formal matrix rings and group rings is discussed in details. A necessary and sufficient condition is obtained for an ideal I of R to have the property that R/I strongly nil-clean implies R is strongly nil-clean. Finally, responding to the question of when a matrix ring is nil-clean, we prove that the matrix ring over a 2-primal ring R is nil-clean iff R/J(R) is boolean and J(R) is nil, i.e., R is strongly nil-clean.