On ideals of triangular matrix rings

We provide a formula for the number of ideals of complete block-triangular matrix rings over any ring R such that the lattice of ideals of R is isomorphic to a finite product of finite chains, as well as for the number of ideals of (not necessarily complete) block-triangular matrix rings over any such ring R with three blocks on the diagonal.     

Study of formal triangular matrix rings

In this paper we carry out a systematic study of various ring theoretic properties of formal triangular matrix rings. Some definitive results are obtained on these rings concerning properties such as being respectively left Kasch, right mininjective, clean, potent, right PF or a ring of stable rank ≤ n. The concepts of a strong left Kasch ring, a strong right mininjective ring are introduced and it is determined when the triangular matrix rings are respectively strong left Kasch or strong right mininjective. It is also proved that being strong left Kasch or strong right mininjective are Morita invariant properties.     

Automorphism groups of generalized triangular matrix rings

We call a ring strongly indecomposable if it cannot be represented as a non-trivial (i.e. M≠0) generalized triangular matrix ring , for some rings R and S and some R-S-bimodule _RM_S. Examples of such rings include rings with only the trivial idempotents 0 and 1, as well as endomorphism rings of vector spaces, or more generally, semiprime indecomposable rings. We show that if R and S are strongly indecomposable rings, then the triangulation of the non-trivial generalized triangular matrix ring is unique up to isomorphism; to be more precise, if is an isomorphism, then there are isomorphisms ρ:R→R^′ and ψ:S→S^′ such that χ:=φ_∣M:M→M^′ is an R-S-bimodule isomorphism relative to ρ and ψ. In particular, this result describes the automorphism groups of such upper triangular matrix rings      

Jordan homomorphisms of upper triangular matrix rings

In this paper we investigate Jordan homomorphisms of upper triangular matrix rings and give a sufficient condition under which they are necessarily homomorphisms or anti-homomorphisms.     

The Auslander-type condition of triangular matrix rings

Let R be a left and right Noetherian ring and n,k be any non-negative integers.R is said to satisfy the Auslander-type condition G n (k) if the right flat dimension of the (i + 1)-th term in a minimal injective resolution of RR is at most i + k for any 0 i n 1.In this paper,we prove that R is Gn (k) if and only if so is a lower triangular matrix ring of any degree t over R.     

Jordan isomorphisms of upper triangular matrix rings

Let R be a 2-torsionfree ring with identity 1 and let T_n(R), n ? 2, be the ring of all upper triangular n × n matrices over R. We describe additive Jordan isomorphisms of T_n(R) onto an arbitrary ring and generalize several results on this line.     

Commuting traces of upper triangular matrix rings

Let \(T_n(R)\) be the upper triangular matrix ring over a unital ring R. Suppose that \(B:T_n(R)\times T_n(R) \rightarrow T_n(R)\) is a biadditive map such that \(B(X,X)X = XB(X,X)\) for all \(X \in T_n(R)\). We consider the problem of describing the form of the map \(X\mapsto B(X,X)\).     

Functional identities in upper triangular matrix rings revisited

The aim of this paper is to give an improvement of a result on functional identities in upper triangular matrix rings obtained by Eremita, which presents a short proof of Eremita’s result.     

Jordan isomorphisms of triangular rings

We investigate Jordan isomorphisms of triangular rings and give a sufficient condition under which they are necessarily isomorphisms or anti-isomorphisms. As corollaries we obtain generalizations of two recent results: the one concerning Jordan isomorphisms of triangular matrix algebras by Beidar, Bresar and Chebotar, and the one concerning Jordan isomorphisms of nest algebras by Lu.

     

Additivity of multiplicative maps on triangular rings

In this paper we shall give a unified technique in the discussion of the additivity of n-multiplicative automorphisms, n-multiplicative derivations, n-elementary surjective maps, and Jordan multiplicative surjective maps on triangular rings.     

Hereditary triangular matrix comonads

We recognize Harada’s generalized categories of diagrams as a particular case of modules over a monad defined on a finite direct product of additive categories. We work in the dual (albeit formally equivalent) situation, that is, with comodules over comonads. With this conceptual tool at hand, we obtain several of the Harada results with simpler proofs, some of them under more general hypothesis, besides with a characterization of the normal triangular matrix comonads that are hereditary, that is, of homological dimension less than or equal to 1. Our methods rest on a matrix representation of additive functors and natural transformations, which allows us to adapt typical algebraic manipulations from Linear Algebra to the additive categorical setting.