Сопряженные идемпотентные формальные матрицы второго порядка над кольцами вычетов

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.

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.

In this chapter, we define formal matrix rings of order 2 and formal matrix rings of arbitrary order n. Their main properties are considered and examples of such rings are given. We indicate the relationship between formal matrix rings, endomorphism rings of modules, and systems of orthogonal idempotents of rings. For formal matrix rings, the Jacobson radical and the prime radical are described. We find when a formal matrix ring is Artinian, Noetherian, regular, unit-regular, and of stable rank 1. In the last section, clean and k-good matrix rings are considered.

A class of formal matrix rings

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.

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.

Let us recall some classes of rings. A ring R is said to be k-nil-clean if each element can be written as a sum of a nilpotent and k idempotents. A ring R is said to be fine if each non-zero element can be written as a sum of a unit and a nilpotent. A ring R is called nil-good if every element is a nilpotent or a sum of a nilpotent and a unit. And, finally, ring R is called nil-good clean if every element is a sum of a nilpotent, an idempotent, and a unit. In this paper, we continue our work on additive problems in formal matrix rings over residue class rings. We have found necessary and sufficient conditions for the nilpotency of a formal matrix over residue class rings. After that we have shown that a ring of such matrices is (p –1)-nil-clean and nil-good clean. Also, answering the question posed in the previous article of the second co-author, we prove that a ring of formal matrices over residue rings is never nil-good, and, therefore, not fine.

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.

A module M is called pseudo-projective if every epimorphism from M to each quotient module of M can be lifted to an endomorphism of M. In this paper, we study some properties of pseudo-projective modules and their endomorphism rings. It shows that if M is a self-cogenerator pseudo-projective module with finite hollow dimension, End(M) is a semilocal ring and every maximal right ideal of End(M) has of the form {s ? End(M)| Im(s) + Ker(h) ? M} for some endomorphism h of M with h(M) hollow. Moreover, it shows that a pseudo-projective R-module Mis an SSP-module if and only if the product of any two regular elements of End(M) is a regular element. Finally, we investigate the pseudo-projectivity of modules over a formal triangular matrix ring.

Nil-clean and strongly nil-clean 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.

Let Xn be a set with finite number of elements following natural ordering of numbers. The formulae for the total number of elements in partial one – one convex and contraction transformation semigroup and its idempotents are obtained and discussed in this paper. The relationship between fix α and idempotency is obtained and stated; an element α is an idempotent if |Imα |= |ƒ (α) |. Also, idempotents commute and the product of two or more idempotents is an idempotent. An element y is said to be unique if x is an invertible element such that xy = yx = 1, x, y S. Key words: Convex, contraction, fix element, idempotent and partial one – one transformation semigroup.

Let S be a formal matrix ring, T the subring consisting of all diagonal elements, I the set consisting of all off-diagonal elements. Then I is a split radical ideal under certain conditions. In this paper, we show that K2(s)≃K2(T)⊕K2(S, I), and a presentation of K2(S, I) is given.

Let p be a prime number, m, n be natural and m n > 0. Let the formal matrix ring (Z/pmZ Z/pnZ Z/pnZ Z/pnZ) be isomorphic to the endomorphism ring E((Z/pmZ) ⊕ (Z/ pnZ)), may be of interest in data encryption. We will show that the ring E((Z/pmZ) ⊕ (Z/pnZ)), m ≥ n, is 2-good and 2-nil-good for p > 2 and not good for p = 2 and m > n .

This work contains some new and known results on modules over formal matrix rings. The main results are presented with proofs.