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.

     

亚直不可约环成为全矩阵环的条件

给出了亚直不可约环是除环上的全矩阵环的一个新的条件。     

On extensions of matrix rings with skew Hochschild 2-cocycles

We study structures of Hochschild 2-cocycles related to endomorphisms and introduce a skew Hochschild 2-cocycle. We moreover define skew Hochschild extensions equipped with skew Hochschild 2-cocycles, and then we examine uniquely clean, Abelian, directly finite, symmetric, and reversible ring properties of skew Hochschild extensions and related ring systems. The results obtained here provide various kinds of examples of such rings. Especially, we give an answer negatively to the question of H. Lin and C. Xi for the corresponding Hochschild extensions of reversible (or semicommutative) rings. Finally, we establish three kinds of Hochschild extensions with Hochschild 2-cocycles and skew Hochschild 2-cocycles.     

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.     

Gorenstein flatness and injectivity over Gorenstein rings

Let R be a Gorenstein ring.We prove that if I is an ideal of R such that R/I is a semi-simple ring,then the Gorenstein flat dimension of R/I as a right R-module and the Gorenstein injective dimension of R/I as a left R-module are identical.In addition,we prove that if R→S is a homomorphism of rings and SE is an injective cogenerator for the category of left S-modules,then the Gorenstein flat dimension of S as a right R-module and the Gorenstein injective dimension of E as a left R-module are identical.We also give some applications of these results.     

Characterizing Cohen-Macaulay local rings by Frobenius maps

Let be a commutative noetherian local ring of prime characteristic. Denote by the ring regarded as an -algebra through -times composition of the Frobenius map. Suppose that is F-finite, i.e., is a finitely generated -module. We prove that is Cohen-Macaulay if and only if the -modules have finite Cohen-Macaulay dimensions for infinitely many integers .

     

三级三角矩阵环上模范畴和同调刻划

设Γ是三级三角矩阵代数,m odΓ表示Γ上的有限生成模范畴,ΓL是与m odΓ等价的范畴.讨论了ΓL的Jacabson根,ΓL的单对象及投射对象的形式及Γ的整体维数等同调性质.     

Additivity of Jordan maps on triangular algebras

The aim of this article is to prove a result on the additivity of Jordan maps on triangular algebras. As a consequence the additivity of Jordan maps on upper triangular matrix algebras over a faithful commutative ring of 2-torsion free is determined.     

Derived equivalences between triangular matrix algebras

In this paper, we study derived equivalences between triangular matrix algebras using certain classical recollements. We show that special properties of these recollements actually characterize triangular matrix algebras and describe methods to construct tilting modules and tilting complexes inducing derived equivalences between them.     

On certain homological finiteness conditions

In this paper, we show that the injective dimension of all projective modules over a countable ring is bounded by the self-injective dimension of the ring. We also examine the extent to which the flat length of all injective modules is bounded by the flat length of an injective cogenerator. To that end, we study the relation between these finiteness conditions on the ring and certain properties of the (strict) Mittag–Le?er modules. We also examine the relation between the self-injective dimension of the integral group ring of a group and Ikenaga’s generalized (co-)homological dimension.     

The weak dimensions of Gaussian rings

We provide necessary and sufficient conditions for a Gaussian ring to be semihereditary, or more generally, of . Investigating the weak global dimension of a Gaussian coherent ring , we show that the only values may take are and ; but that is always at most one. In particular, we conclude that a Gaussian coherent ring is either Von Neumann regular, or semihereditary, or non-regular of .

     

Normal dilatation of triangular matrices

LetR be a (real or complex) triangular matrix of ordern, say, an upper triangular matrix. Is it true that there exists a normaln×n matrixA whose upper triangle coincides with the upper triangle ofR? The answer to this question is “yes” and is obvious in the following cases: (1)R is real; (2)R is a complex matrix with a real or a pure imaginary main diagonal, and moreover, all the diagonal entries ofR belong to a straight line. The answer is also in the affirmative (although it is not so obvious) for any matrixR of order 2. However, even forn=3 this problem remains unsolved. In this paper it is shown that the answer is in the affirmative also for 3×3 matrices.     

Jordan isomorphisms of generalized structural matrix rings

We describe the sub-bimodules of matrix bimodules over two structural matrix rings. Structural matrix bimodules arise as particular such sub-bimodules, and we discuss when such a bimodule is faithful or indecomposable. As an application, we obtain a large class of rings whose Jordan isomorphisms are either ring isomorphisms or ring anti-isomorphisms. Complete upper block triangular matrix rings over 2-torsion-free indecomposable rings are elements of this class.     

Square-zero derivations of matrix algebras over commutative rings

Let R be a commutative ring with identity, M _n (R) the R-algebra consisting of all n by n matrices over R. In this article, for n ≥ 5 we classify linear maps φ from M _n (R) into itself satisfying φ(x)x + xφ(x) = 0 whenever x ² = 0. We call such maps as square-zero derivations.     

环的Extensions扩张与余纯维数(英文)

Let R be a noetherian ring and S an excellent extension of R.cid(M) denotes the copure injective dimension of M and cfd(M) denotes the copure flat dimension of M.We prove that if M S is a right S-module then cid(M S)=cid(M R) and if S M is a left S-module then cfd(S M)=cfd(R M).Moreover,cid-D(S)=cid-D(R) and cfd-D(S)=cfdD(R).