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 $R⋉M$ be a trivial extension of a ring $R$ by an $R$-$R$-bimodule $M$. We first study the properties of pure projective and pure injective modules over $R⋉M$. Then we characterize $FP$-injective modules over $R⋉M$. Finally some applications are given to Morita context rings.

Given a finite-dimensional monomial algebra A, we consider the trivial extension TA and provide formulae, depending on the characteristic of the field, for the dimensions of the summands HH₁(A) and Alt(DA) of the first Hochschild cohomology group HH¹(TA). From these a formula for the dimension of HH¹(TA) can be derived.

Given a ring R, we study the bimodules M for which the trivial extension R ∝ M is morphic. We obtain a complete characterization in the case where R is left perfect, and we prove that R ∝ Q/R is morphic when R is a commutative reduced ring with classical ring of quotients Q. We also extend some known results concerning the connection between morphic rings and unit regular rings.

We study the complexity of a family of finite-dimensional self-injective k-algebras where k is an algebraically closed field. More precisely, let T be the trivial extension of an iterated tilted algebra of type H. We prove that modules over the trivial extension T all have complexities either 0, 1, 2 or infinity, depending on the representation type of the hereditary algebra H.

In this note, a necessary and sufficient condition for a trivial extension to be a left p.p.-ring is obtained. As an application, we determine when a trivial Morita context is a left p.p.-ring, and a result of Xue [On p.p. rings, Kobe J. Math. 7 (1990) 77–80] characterizing triangular p.p.-rings follows immediately.

Given a semidualizing module $C$ over a commutative Noetherian ring, Holm and Jørgensen [Semi-dualizing modules and related Gorenstein homological dimensions, J. Pure Appl. Algebra205(2) (2006) 423–445] investigate some connections between $C$-Gorenstein dimensions of an $R$-complex and Gorenstein dimensions of the same complex viewed as a complex over the “trivial extension” $R⋉C$. We generalize some of their results to a certain type of retract diagram. We also investigate some examples of such retract diagrams, namely D’Anna and Fontana’s amalgamated duplication [An amalgamated duplication of a ring along an ideal: The basic properties, J. Algebra Appl.6(3) (2007) 443–459] and Enescu’s pseudocanonical cover [A finiteness condition on local cohomology in positive characteristic, J. Pure Appl. Algebra216(1) (2012) 115–118].

Recently, Xiang and Ouyang defined a (commutative unital) ring $R$ to be Nil${}_{*}$-coherent if each finitely generated ideal of $R$ that is contained in Nil$(R)$ is a finitely presented $R$-module. We define and study Nil${}_{*}$-coherent modules and special Nil${}_{*}$-coherent modules over any ring. These properties are characterized and their basic properties are established. Any coherent ring is a special Nil${}_{*}$-coherent ring and any special Nil${}_{*}$-coherent ring is a Nil${}_{*}$-coherent ring, but neither of these statements has a valid converse. Any reduced ring is a special Nil${}_{*}$-coherent ring (regardless of whether it is coherent). Several examples of Nil${}_{*}$-coherent rings that are not special Nil${}_{*}$-coherent rings are obtained as byproducts of our study of the transfer of the Nil${}_{*}$-coherent and the special Nil${}_{*}$-coherent properties to trivial ring extensions and amalgamated algebras.

Our goal is to determine when the trivial extensions of commutative rings by modules are Cohen–Macaulay in the sense of Hamilton and Marley. For this purpose, we provide a generalization of the concept of Cohen–Macaulayness of rings to modules.

Let $R$ be a commutative ring with $1\ne 0$. Recall that a proper ideal $I$ of $R$ is called a 2-absorbing ideal of $R$ if $a,b,c\in R$ and $abc\in I$, then $ab\in I$ or $ac\in I$ or $bc\in I$. A more general concept than 2-absorbing ideals is the concept of $n$-absorbing ideals. Let $n\ge 1$ be a positive integer. A proper ideal $I$ of $R$ is called an n-absorbing ideal of $R$ if $a_1,a_2,\dots ,a_{n+1}\in R$ and $a_1,a_2\cdots a_{n+1}\in I$, then there are $n$ of the $a_i$’s whose product is in $I$. The concept of $n$-absorbing ideals is a generalization of the concept of prime ideals (note that a prime ideal of $R$ is a 1-absorbing ideal of $R$). Let $m$ and $n$ be integers with $1\le n<m$. A proper ideal $I$ of $R$ is called an $(m,n)$-closed ideal of $R$ if whenever $a^m\in I$ for some $a\in R$ implies $a^n\in I$. Let $A$ be a commutative ring with $1\ne 0$ and $M$ be an $A$-module. In this paper, we study $n$-absorbing ideals and $(m,n)$-closed ideals in the trivial ring extension of $A$ by $M$ (or idealization of $M$ over $A$) that is denoted by $A(+)M$.

Let $A\subseteq B$ be a commutative ring extension such that $B$ is a trivial extension of $A$ (denoted by $A\propto E$) or an amalgamated duplication of $A$ along some ideal of $A$ (denoted by $A\bowtie I)$. This paper examines the transfer of AM-ring, N-ring, SSP-ring and SP-ring between $A$ and $B$. We study the transfer of those properties to trivial ring extension. Call a special SSP-ring an SSP-ring of the following type: it is the trivial extension of $B\times C$ by a C-module $E$, where $B$ is an SSP-ring, $C$ a von Neumann regular ring and $E$ a multiplication C-module. We show that every SSP-ring with finitely many minimal primes which is a trivial extension is in fact special. Furthermore, we study the transfer of the above properties to amalgamated duplication along an ideal with some extra hypothesis. Our results allows us to construct nontrivial and original examples of rings satisfying the above properties.

Local dimension is an ordinal valued invariant that is in some sense a measure of how far a ring is from being local and denoted $\mathrm{\text{ldim}}(R)$. The purpose of this paper is to study the local dimension of ring extensions such as homomorphic image, trivial ring extension and the amalgamation of rings.

Let $R⋉M$ be a trivial extension of a ring $R$ by an $R$-$R$-bimodule $M$. We first study the transfer of some rings such as left $V$-rings, left Kasch rings, left PS rings between $R$ and $R⋉M$. Then we investigate how to construct $n$-tilting modules and Gorenstein modules in $R⋉M$-Mod from $R$-Mod. Finally we exhibit the connections between preenvelopes (respectively, precovers) in $R$-Mod and preenvelopes (respectively, precovers) in $R⋉M$-Mod.

We introduce the concepts of generalized compatible and cocompatible bimodules in order to characterize Gorenstein projective, injective and flat modules over trivial ring extensions. Let $R⋉M$ be a trivial extension of a ring $R$ by an $R$-$R$-bimodule $M$ such that $M$ is a generalized compatible $R$-$R$-bimodule and $Z(R)$ is a generalized compatible $R⋉M$-$R⋉M$-bimodule. We prove that $(X,\alpha )$ is a Gorenstein projective left $R⋉M$-module if and only if the sequence $M{\otimes }_{R}M{\otimes }_{R}X\stackrel{M\otimes \alpha }{\to }M{\otimes }_{R}X\stackrel{\alpha }{\to }X$ is exact and coker($\alpha$) is a Gorenstein projective left $R$-module. Analogously, we explicitly characterize Gorenstein injective and flat modules over trivial ring extensions. As an application, we describe Gorenstein projective, injective and flat modules over Morita context rings with zero bimodule homomorphisms.

We introduce the notion of an α-reflexive ring to extend the concept of a reflexive ring and that of an α-rigid ring. We first consider some basic properties of α-reflexive rings, including some examples needed in the process. We prove that a ring R is α-rigid if and only if R is a reduced α-reflexive ring with α a monomorphism. We next investigate the α-reflexivity of some kinds of polynomial rings. It is shown that if R is a reduced α-reflexive ring with α(1) = 1, then R[x]/(xⁿ) is an α-reflexive ring, where (xⁿ) is the ideal generated by xⁿ. Moreover, for an Armendariz ring R, we prove that R is α-reflexive if and only if R[x] is α-reflexive if and only if R[x; x^-1] is α-reflexive. As a sequence, some known results relating to reflexive rings are obtained as corollaries of these results.

The concept of central reflexive rings is introduced in this paper as a generalization of reflexive rings. Since every reflexive ring is central reflexive, we study the sufficient condition for a central reflexive ring to be a reflexive one. It is shown that every central reversible and hence every central symmetric ring is central reflexive, however converse implications are wrong. It is also proven that the polynomial ring R[x] is central reflexive if R is central reflexive and quasi-Armendariz. We have given some characterizations for an α-reflexive ring and also improved some of the results given by Zhao and Zhu [Extensions of α-reflexive rings, Asian-European J. Math.5(1) (2012) 1–10]. Finally we introduced the concept of central α-reflexive rings as a generalization of α-reflexive rings.

Mason introduced the notion of reflexive property for rings which play roles in noncommutative ring theory. In this paper, we extend this property to rings with involution and investigate their properties. We provide many examples of these rings and also consider some extensions such as trivial extension, Dorroh extension, etc.