The classes of FP-injective and weakly quasi-Frobenius rings are investigated. The properties for both classes of rings are closely linked with embedding of finitely presented modules in fp-flat and free modules respectively. Using these properties, we describe the classes of coherent CF and FGF-rings. Moreover, it is proved that the group ring R(G) is FP-injective (resp. weakly quasi-Frobenius) if and only if the ring R is FP-injective (resp. weakly quasi-Frobenius) and the group G is locally finite.
It is not known whether right CF-rings (FGF-rings) are right artinian (quasi-Frobenius). This paper gives a positive answer of this question in the case of weak CS (s-CS) and GC2 rings. Also we get some new results on almost injective rings.
We develop a theory of \emph{locally Frobenius algebras} which are colimits of certain directed systems of Frobenius algebras. A major goal is to obtain analogues of the work of Moore \& Peterson and Margolis on \emph{nearly Frobenius algebras} and \emph{$P$-algebras} which was applied to graded Hopf algebras such as the Steenrod algebra for a prime. Such locally Frobenius algebras are coherent and in studying their modules we are naturally led to focus on coherent and finite dimensional modules. Indeed, the category of coherent modules over locally Frobenius algebra $A$ is abelian with enough…
We introduce and investigate ss-injectivity as a generalization of both soc-injectivity and small injectivity. A module M is said to be ss-N-injective (where N is a module) if every R-homomorphism from a semisimple small submodule of N into M extends to N. A module M is said to be ss-injective (resp. strongly ss-injective), if M is ss-R-injective (resp. ss-N-injective for every right R-module N). Some characterizations and properties of (strongly) ss-injective modules and rings are given. Some results of Amin, Yuosif and Zeyada on soc-injectivity are extended to ss-injectivity. Also, we provid…
The Faith-Menal conjecture says that every strongly right $Johns$ ring is $QF$. The conjecture is also equivalent to say every right noetherian left $FP$-injective ring is $QF$. In this short article, we show that the conjecture is true under the condition(a proper generalization of left $CS$ condition)that every nonzero complement left ideal is not small(a left ideal $I$ is called small if for every left ideal $K$, $K$+$I$=$R$ implies $K$=$R$). It is also proved that (1) $R$ is $QF$ if and only if $R$ is a left and right mininjective ring with $ACC$ on right annihilators in which $S_{r}\subse…
Preenvelopes of acts over a monoid are defined by analogy with Enochs' definition of preenvelopes of modules. Provided that it is closed for pure subacts, a class of acts is shown to be preenveloping precisely when it is closed under direct products. Examples of such classes include absolutely pure, weakly f-injective and weakly p-injective acts.
MacWilliams proved that every finite field has the extension property for Hamming weight which was later extended in a seminal work by Wood who characterized finite Frobenius rings as precisely those rings which satisfy the MacWilliams extension property. In this paper, the question of when is a MacWilliams ring quasi-Frobenius is addressed. It is proved that a right or left noetherian left 1-MacWilliams ring is quasi-Frobenius thus answering the different questions asked in [M. C. Iovanov, On infinite MacWilliams rings and minimal injectivity conditions, Proc. Amer. Math. Soc., DOI: 10.1090/p…
Given a morphism of (small) groupoids with injective object map, we provide sufficient and necessary conditions under which the induction and co-induction functors between the categories of linear representations are naturally isomorphic. A morphism with this property is termed a Frobenius morphism of groupoids. As a consequence, an extension by a subgroupoid is Frobenius if and only if each fibre of the (left or right) pull-back biset has finitely many orbits. Our results extend and clarify the classical Frobenius reciprocity formulae in the theory of finite groups, and characterize Frobenius…
A partition of the set of unary pp formulas into four regions is presented, which has a bearing on various structural properties of modules. The machinery developed allows for applications to IF, weakly coherent, nonsingular, and reduced rings, as well as domains, specifically Ore domains. One of the four types of formula are called high. These are used to define Ulm submodules and Ulm length of modules over any associative ring. It is shown that pure injective modules have Ulm length at most 1. As a consequence, pure injective modules over RD domains (in particular, pure injective modules ove…
Let $\mathbf{k}$ be an algebraically closed field of arbitrary characteristic, let $Λ$ be a finite dimensional $\mathbf{k}$-algebra and let $V$ be a $Λ$-module with stable endomorphism ring isomorphic to $\mathbf{k}$. If $Λ$ is self-injective, then $V$ has a universal deformation ring $R(Λ,V)$, which is a complete local commutative Noetherian $\mathbf{k}$-algebra with residue field $\mathbf{k}$. Moreover, if $Λ$ is further a Frobenius $\mathbf{k}$-algebra, then $R(Λ,V)$ is stable under syzygies. We use these facts to determine the universal deformation rings of string $Λ_{m,N}$-modules whose c…
For any ring R we construct two triangulated categories, each admitting a functor from R-modules that sends projective and injective modules to 0. When R is a quasi-Frobenius or Gorenstein ring, these triangulated categories agree with each other and with the usual stable module category. Our stable module categories are homotopy categories of Quillen model structures on the category of R-modules. These model categories involve generalizations of Gorenstein projective and injective modules that we derive by replacing finitely presented modules by modules of type FP-infinity. Along the way, we …
In this paper we study initial topological properties of the (non-)finitely-generated locus of Frobenius Algebra coming from Stanley-Reisner rings defined through face ideals. More specifically, we will give a partial answer to a conjecture made by M. Katzman about the openness of the finitely generated locus of such Frobenius algebras. This conjecture can be formulated in a precise manner: Let us define \begin{equation*} U=\{P\in \text{Spec}(R)~:~\mathcal{F}(E_{R_{p}})~\text{is a finitely generated}~R_{p}\text{-algebra}\} \end{equation*} where $\mathcal{F}$ denotes the Frobenius functor and $…
By a map $p:Q\to X$ of involutive quantales is meant a homomorphism $p^*:X\to Q$. Calling a map $p$ weakly open if $p^*$ has a left adjoint $p_!$ which satisfies the Frobenius reciprocity condition (i.e., $p_!$ is a homomorphism of $X$-modules), we say that $p$ is open if it is stably weakly open. We also study a two-sided version, FR2, of the Frobenius reciprocity condition, and show that the weakly open surjections that satisfy FR2 are open. Maps of the latter kind arise in the study of Fell bundles on groupoids.
Finite Frobenius rings have been characterized as precisely those finite rings satisfying the MacWilliams extension property, by work of Wood. In the present note we offer a generalization of this remarkable result to the realm of Artinian rings. Namely, we prove that a left Artinian ring has the left MacWilliams property if and only if it is left pseudo-injective and its finitary left socle embeds into the semisimple quotient. Providing a topological perspective on the MacWilliams property, we also show that the finitary left socle of a left Artinian ring embeds into the semisimple quotient i…
In this paper, we introduce the classes of weakly surjunctive and linearly surjunctive groups which include all sofic groups and more generally all surjunctive groups. We investigate various properties of such groups and establish in particular a reversibility and invertibility theorem for injective endomorphisms of symbolic group varieties over weakly surjunctive group universes with algebraic group alphabets in arbitrary characteristic. We also obtain novel evidence related to Kaplansky's stable finiteness conjecture.
In this paper, we introduce the class of Cohen-Macaulay (=CM) dg (=differential graded) modules over Gorenstein dg algebras and study their basic properties. We show that the category of CM dg modules forms a Frobenius extriangulated category, in the sense of Nakaoka and Palu, and it admits almost split extensions. We also study representation-finite $d$-self-injective dg algebras $A$ in detail. In particular, we classify the Auslander-Reiten (=AR) quivers of CM $A$ for those $A$ in terms of $(-d)$-Calabi-Yau (=CY) configurations, which are Riedtmann's configuration for the case $d=1$. For any…
