Narrow Results

We study (support) $\tau$-tilting modules over the trivial extensions of finite-dimensional algebras. More precisely, we construct two classes of (support)$\tau$-tilting modules in terms of the adjoint functors which extend and generalize the results on (support) $\tau$-tilting modules over triangular matrix rings given by Gao-Huang.

Let A and B be rings and $T=\left(\begin{array}{cc}\hfill A\hfill & \hfill M\hfill \\ \hfill 0\hfill & \hfill B\hfill \end{array}\right)$ with M an A-B-bimodule. We first construct a semi-complete duality pair ${{𝒟}}_T$ over T using duality pairs over A and B, respectively. Then we characterize when a left T-module is Gorenstein projective, injective or flat with respect to the duality pair ${{𝒟}}_T$. As applications, we investigate when a left T-module is projectively coresolved Gorenstein flat, Ding injective or Gorenstein flat. In addition, we give an estimate of Gorenstein flat dimension of a left T-module with respect to ${{𝒟}}_T$. Finally, we construct model structures and recollements relative to these generalized Gorenstein modules.

We determine conditions under which a generalized triangular matrix ring has finite reduced rank, in the general torsion-theoretic sense. These are applied to characterize certain orders in Artinian rings, and to show that if each homomorphic image of a ring S has finite reduced rank, then so does the ring of lower triangular matrices over S.

Let $A$ and $B$ be rings, $U$ a $(B,A)$-bimodule and $T=\left(\right.\begin{array}{cc}\hfill A\hfill & \hfill 0\hfill \\ \hfill U\hfill & \hfill B\hfill \end{array}\left)\right.$ be a triangular matrix ring. We first give an explicit description for Gorenstein injective $T$-modules over the triangular matrix ring $T$, and then construct a (right) recollement of stable categories of Gorenstein injective modules.

Let R be a ring with an endomorphism σ. We show that the clean property and various Armendariz-type properties of R are inherited by the skew matrix rings S(R,n,σ) and T(R,n,σ). They allow the construction of rings with a non-zero nilpotent ideal of arbitrary index of nilpotency which inherit various interesting properties of rings.

A generalization of semiprime rings and right p.q.-Baer rings, which we call quasi-Armendariz rings of differential inverse power series type (or simply, $\mathcal{D}\mathcal{I}\mathcal{P}\mathcal{S}$-quasi-Armendariz), is introduced and studied. It is shown that the $\mathcal{D}\mathcal{I}\mathcal{P}\mathcal{S}$-quasi-Armendariz rings are closed under direct sums, upper triangular matrix rings, full matrix rings and Morita invariance. Various classes of non-semiprime $\mathcal{D}\mathcal{I}\mathcal{P}\mathcal{S}$-quasi-Armendariz rings are provided, and a number of properties of this generalization are established. Some characterizations for the differential inverse power series ring $R[[x^{-1};\delta ]]$ to be quasi-Baer, generalized quasi-Baer, primary, nilary, reflexive, ideal-symmetric and left AIP are concluded, where δ is a derivation on the ring R. Finally, miscellaneous examples to illustrate and delimit the theory are given.

In this paper, we study various annihilator properties in a ring $R$ with an endomorphism $\sigma$ and some subrings of the skew triangular matrix rings $T_n(R,\sigma )$. They allow the construction of rings with a nonzero nilpotent ideal of arbitrary index of nilpotency which have various zero-divisor properties.

For a ring endomorphism $\alpha$, a generalization of semiprime rings and right p.q.-Baer rings, which we call quasi-Armendariz rings of skew Hurwitz series type (or simply, $\alpha$-${𝒮}{\mathscr{H}}{𝒬}{𝒜}$), is introduced and studied. It is shown that the ${𝒮}{\mathscr{H}}{𝒬}{𝒜}$-rings are closed upper triangular matrix rings, full matrix rings and Morita invariance. Some characterizations for the skew Hurwitz series ring $(HR,\alpha )$ to be quasi-Baer, generalized quasi-Baer, primary, nilary, reflexive, ideal-symmetric and semiprime are concluded.