The early papers of Hyman Bass in the late 50s and the early 60s leading up to his pioneering work in algebraic K-theory have played an important and very special role in ring theory and the theory of projective (and injective) modules. In this article, we give a general survey of Bass's fundamental contributions in this early period of his work, and explain how much this work has influenced and shaped the thinking of subsequent researchers in the area.
We give a constructive proof of the theorem of Bass and Milnor saying that if $G$ is a finite abelian group then the Bass units of the integral group ring $\Z G$ generate a subgroup of finite index in its units group $\U(\Z G)$. Our proof provides algorithms to represent some units that contribute to only one simple component of $\Q G$ and generate a subgroup of finite index in $\U(\Z G)$ as product of Bass units. We also obtain a basis $B$ formed by Bass units of a free abelian subgroup of finite index in $\U(\Z G)$ and give, for an arbitrary Bass unit $b$, an algorithm to express $b^{φ(|G|)}…
Hereditarily and Super Bassian Modules over Certain Rings
Bulgarian Academy of Sciences, Institute of Mathematics and Informatics（保加利亚科学院数学与信息学研究所）；Faculty of Mathematics, The University of Danang - University of Science and Education（岘港大学-科学教育大学数学学院）；Department of Algebra, MFF UK（查理大学数学与物理学院代数系）
It is proved that the exchange property, the Bass stable rank and the quasi-Bass property are all preserved under surjective inverse limits. This is then applied to multiplier rings by showing that in many cases can be obtained as inverse limits.
In this survey we determine an explicit set of generators of the maximal ideals in the ring $\mathbb R[x_1,\dots,x_n]$ of polynomials in $n$ variables with real coefficients and give an easy analytic proof of the Bass-Vasershtein theorem on the Bass stable rank of $\mathbb R[x_1,\dots,x_n]$. The ingredients of the proof stem from different publications by Coquand, Lombardi, Estes and Ohm. We conclude with a calculation of the topological stable rank of $\mathbb R[x_1,\dots,x_n]$, which seems to be unknown so far.
We extend the definition of a semidualizing module to associative rings. This enables us to define and study Auslander and Bass classes with respect to a semidualizing bimodule C. We then study the classes of C-flats, C-projectives, and C-injectives, and use them to provide a characterization of the modules in the Auslander and Bass classes. We extend Foxby equivalence to this new setting. This paper contains a few results which are new in the commutative, noetherian setting.
The Bohl algebra $\textrm{B}$ is the ring of linear combinations of functions $t^k e^{λt}$, where $k$ is any nonnegative integer, and $λ$ is any complex number, with pointwise operations. We show that the Bass stable rank and the topological stable rank of $\textrm{B}$ (where we use the topology of uniform convergence) are infinite.
An element $a$ in a ring $R$ is strongly J-clean if it is the sum of an idempotent and an element in the Jacobson radical that commutes. We characterize the strongly J-clean $2\times 2$ matrices over 2-projective-free non-commutative rings.
An element of a ring $R$ is called strongly $J^{\#}$-clean provided that it can be written as the sum of an idempotent and an element in $J^{\#}(R)$ that commute. We characterize, in this article, the strongly $J^{\#}$-cleanness of matrices over projective-free rings. These extend many known results on strongly clean matrices over commutative local rings.
We prove that every non-finitely generated projective module over the integral group ring of a polycyclic-by-finite group G is free if and only if G is polycyclic.
We discuss the projective line $\mathbb{P}(R)$ over a finite associative ring with unity. $\mathbb{P}(R)$ is naturally endowed with the symmetric and anti-reflexive relation "distant". We study the graph of this relation on $\mathbb{P}(R)$ and classify up to isomorphism all distant graphs $G(R, Δ)$ for rings $R$ up to order $p^5$, $p$ prime.
We study some properties of graded idealizer rings with an emphasis on applications to the theory of noncommutative projective geometry. In particular we give examples of rings for which the $χ$-conditions of Artin and Zhang and the strong noetherian property have very different behavior on the left and right sides.
Let $R$ be any ring with identity and Ch($R$) the category of chain complexes of (left) $R$-modules. We show that the Gorenstein AC-projective chain complexes are the cofibrant objects of an abelian model structure on Ch($R$). The model structure is cofibrantly generated and is projective in the sense that the trivially cofibrant objects are the categorically projective chain complexes. We show that when $R$ is a Ding-Chen ring, that is, a two-sided coherent ring with finite self FP-injective dimension, then the model structure is finitely generated, and so its homotopy category is compactly g…
Recently, in a series of papers "simple" versions of direct-injective and direct-projective modules have been investigated. These modules are termed as "simple-direct-injective" and "simple-direct-projective", respectively. In this paper, we give a complete characterization of the aforementioned modules over the ring of integers and over semilocal rings. The ring is semilocal if and only if every right module with zero Jacobson radical is simple-direct-projective. The rings whose simple-direct-injective right modules are simple-direct-projective are fully characterized. These are exactly the l…
Let $R$ be a valuation ring and let $Q$ be its total quotient ring. It is proved that any singly projective (respectively flat) module is finitely projective if and only if $Q$ is maximal (respectively artinian). It is shown that each singly projective module is a content module if and only if any non-unit of $R$ is a zero-divisor and that each singly projective module is locally projective if and only if $R$ is self injective. Moreover, $R$ is maximal if and only if each singly projective module is separable, if and only if any flat content module is locally projective. Necessary and sufficie…
R is called a right WV -ring if each simple right R-module is injective relative to proper cyclics. If R is a right WV -ring, then R is right uniform or a right V -ring. It is shown that for a right WV-ring R, R is right noetherian if and only if each right cyclic module is a direct sum of a projective module and a CS or noetherian module. For a finitely generated module M with projective socle over a V -ring R such that every subfactor of M is a direct sum of a projective module and a CS or noetherian module, we show M = X \oplus T, where X is semisimple and T is noetherian with zero socle. I…
Let $S$ and $R$ be rings and $_SC_R$ a (faithfully) semidualizing bimodule. We introduce and study $C$-weak flat and $C$-weak injective modules as a generalization of $C$-flat and $C$-injective modules (J. Math. Kyoto Univ. 47(2007), 781-808) respectively, and use them to provide additional information concerning the important Foxby equivalence between the subclasses of the Auslander class $\mathcal{A}_C(R)$ and that of the Bass class $\mathcal{B}_C(S)$. Then we study the stability of Auslander and Bass classes, which enables us to give some alternative characterizations of the modules in $\ma…
We give an explicit description for a basis of a subgroup of finite index in the group of central units of the integral group ring $\Z G$ of a finite abelian-by-supersolvable group such that every cyclic subgroup of order not a divisor of 4 or 6 is subnormal in $G$. The basis elements turn out to be a natural product of conjugates of Bass units. This extends and generalizes a result of Jespers, Parmenter and Sehgal showing that the Bass units generate a subgroup of finite index in the center $\mathcal{Z} (\U (\Z G))$ of the unit group $\U (\Z G)$ in case $G$ is a finite nilpotent group. Next, …
Let $T=\left( \begin{array}{cc} R & M 0 & S \end{array} \right) $ be a triangular matrix ring with $R$ and $S$ rings and $_RM_S$ an $R$-$S$-bimodule. We describe Gorenstein projective modules over $T$. In particular, we refine a result of Enochs, Cortés-Izurdiaga and Torrecillas [Gorenstein conditions over triangular matrix rings, J. Pure Appl. Algebra 218 (2014), no. 8, 1544-1554]. Also, we consider when the recollement of $\mathbb{D}^b(T{\text-} Mod)$ restricts to a recollement of its subcategory $\mathbb{D}^b(T{\text-} Mod)_{fgp}$ consisting of complexes with finite Gorenstein projective di…
Let $R$ be a ring. $R$ is called a right countably $Σ$-C2 ring if every countable direct sum copies of $R_{R}$ is a C2 module. The following are equivalent for a ring $R$: (1) $R$ is a right countably $Σ$-C2 ring. (2) The column finite matrix ring $\mathbb{C}\mathbb{F}\mathbb{M}_{\mathbb{N}}(R)$ is a right C2 (or C3) ring. (3) Every countable direct sum copies of $R_{R}$ is a C3 module. (4) Every projective right $R$-module is a C2 (or C3) module. (5) $R$ is a right perfect ring and every finite direct sum copies of $R_{R}$ is a C2 (or C3) module. This shows that right countably $Σ$-C2 rings a…
We describe rings over which every right module is almost injective. We give a description of rings over which every simple module is a almost projective.
A right $R$-module $M$ is called max-projective provided that each homomorphism $f:M \to R/I$ where $I$ is any maximal right ideal, factors through the canonical projection $π: R \to R/I$. We call a ring $R$ right almost-$QF$ (resp. right max-$QF$) if every injective right $R$-module is $R$-projective (resp. max-projective). This paper attempts to understand the class of right almost-$QF$ (resp. right max-$QF$) rings. Among other results, we prove that a right Hereditary right Noetherian ring $R$ is right almost-$QF$ if and only if $R$ is right max-$QF$ if and only if $R=S\times T$ , where $S$…
Let $M$ be a left module over a ring $R$ and $I$ an ideal of $R$. We call $(P, f)$ a (locally)projective $I$-cover of $M$ if $f$ is an epimorphism from $P$ to $M$, $P$ is (locally)projective, $Kerf\subseteq IP$, and whenever $P=Kerf+X$, then there is a projective summand $Y$ of $P$ in $Kerf$ such that $P=Y\oplus X$. This definition generalizes (locally)projective covers. We characterize $I$-semiregular and $I$-semiperfect rings which are defined by Yousif and Zhou [19] using (locally)projective $I$-covers in section 2 and 3. $I$-semiregular and $I$-semiperfect rings are characterized by projec…
In this paper, the notion of strongly G_C-projective and injective modules is introduced, where C is a semidualizing module. Using these modules we can obtain a new characterization of G_C-projective and injective modules, similar to the one of projective modules by the free modules. We then define and study the global dimensions of rings relative to a semidualizing module C, and prove that the global G_C-projective dimension of a ring R is equal to the global G_C-injective dimension of R.
Let $T_n(q)$ be the ring of lower triangular matrices of order $n \geq 2$ with entries from the finite field $F(q)$ of order $q \geq 2$ and let ${^2T_n(q)}$ denote its free left module. For $n=2,3$ it is shown that the projective line over $T_n(q)$ gives rise to a set of $(q+1)^{(n-1)}q^{\frac{3(n-1)(n-2)}{2}}$ affine planes of order $q$. The points of such an affine plane are non-free cyclic submodules of ${^2T_n(q)}$ not contained in any non-unimodular free cyclic submodule of ${^2T_n(q)}$ and its lines are points of the projective line. Furthermore, it is demonstrated that each affine plane…
We prove that for a noetherian semilocal ring $R$ with exactly $k$ isomorphism classes of simple right modules the monoid $V^*(R)$ of isomorphism classes of countably generated projective right (left) modules, viewed as a submonoid of $V^*(R/J(R))$, is isomorphic to the monoid of solutions in $(\No \cup\{\infty\})^k$ of a system consisting of congruences and diophantine linear equations. The converse also holds, that is, if $M$ is a submonoid of $(\No \cup\{\infty\})^k$ containing an order unit $(n_1,..., n_k)$ of $\No^k$ which is the set of solutions of a system of congruences and linear diop…
We give necessary and sufficient conditions in order for the class of projectively coresolved Gorenstein flat modules, $\mathcal{PGF}$, (respectively that of projectively coresolved Gorenstein $\mathcal{B}$ flat modules, $\mathcal{PGF}_{\mathcal{B}}$) to coincide with the class of Ding projective modules ($\mathcal{DP})$. We show that $\mathcal{PGF} = \mathcal{DP}$ if and only if every Ding projective module is Gorenstein flat. This is the case if the ring $R$ is coherent for example. We include an example to show that the coherence is a sufficient, but not a necessary condition in order to ha…
In the past 15 years a study of ``noncommutative projective geometry'' has flourished. By using and generalizing techniques of commutative projective geometry, one can study certain noncommutative graded rings and obtain results for which no purely algebraic proof is known. For instance, noncommutative graded domains of quadratic growth, or ``noncommutative curves,'' have now been classified by geometric data and these rings must be noetherian. Rings of cubic growth, or ``noncommutative surfaces,'' are not yet classified, but a rich theory is currently forming. In this survey, we describe some…
This paper studies the multiplicative ideal structure of commutative rings in which every finitely generated ideal is quasi-projective. Section 2 provides some preliminaries on quasi-projective modules over commutative rings. Section 3 investigates the correlation with well-known Prufer conditions; namely, we prove that this class of rings stands strictly between the two classes of arithmetical rings and Gaussian rings. Thereby, we generalize Osofsky's theorem on the weak global dimension of arithmetical rings and partially resolve Bazzoni-Glaz's related conjecture on Gaussian rings. We also e…
Let $R$ be a ring and $S$ a multiplicative subset of $R$. An $R$-module $P$ is called uniformly $S$-projective provided that the induced sequence $0\rightarrow \mathrm{Hom}_R(P,A)\rightarrow \mathrm{Hom}_R(P,B)\rightarrow \mathrm{Hom}_R(P,C)\rightarrow 0$ is $u$-$S$-exact for any $u$-$S$-short exact sequence $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$. Some characterizations and properties of $u$-$S$-projective modules are obtained. The notion of $u$-$S$-semisimple modules is also introduced. A ring $R$ is called a $u$-$S$-semisimple ring provided that any free $R$-module is $u$-$S…
Under semi-weak and weak compatibility of bimodules, we establish sufficient and necessary conditions of Gorenstein-projective modules over rings of Morita contexts with one bimodule homomorphism zero. This generalises and extends results on triangular matrix Artin algebras and on Artin algebras of Morita contexts with two bimodule homomorphisms zero in the literature, where only sufficient conditions are given under a strong assumption of compatibility of bimodules. An application is provided to describe Gorenstein-projective modules over noncommutative tensor products arising from Morita con…
We use pullbacks of rings to realize the submonoids $M$ of $(\N_0\cup\{\infty\})^k$ which are the set of solutions of a finite system of linear diophantine inequalities as the monoid of isomorphism classes of countably generated projective right $R$-modules over a suitable semilocal ring. For these rings, the behavior of countably generated projective left $R$-modules is determined by the monoid $D(M)$ defined by reversing the inequalities determining the monoid $M$. These two monoids are not isomorphic in general. As a consequence of our results we show that there are semilocal rings such tha…
We apply the theory of cotorsion pairs to study closure properties of classes of modules with finite projective dimension with respect to direct limit operations and to filtrations. We also prove that if the ring is an order in an $\aleph_0$-noetherian ring Q of small finitistic dimension 0, then the cotorsion pair generated by the modules of projective dimension at most one is of finite type if and only if Q has big finitistic dimension 0. This applies, for example, to semiprime Goldie rings and Cohen Macaulay noetherian commutative rings.
The Hilbert scheme of point modules was introduced by Artin-Tate-Van den Bergh to study non-commutative graded algebras. The key tool is the construction of a map from the algebra to a twisted ring on this Hilbert scheme. In this paper, we study moduli stacks of more general "fat" point modules, and show that there is a similar map to a twisted ring associated to the stack. This is used to provide a sufficient criterion for a non-commutative projective surface to be birationally PI. It is hoped that such a criterion will be useful in understanding Mike Artin's conjecture on the birational clas…
We consider a finite acyclic quiver $\mathcal{Q}$ and a quasi-Frobenius ring $R$. We endow the category of quiver representations over $R$ with a model structure, whose homotopy category is equivalent to the stable category of Gorenstein-projective modules over the path algebra $R\mathcal{Q}$. As an application, we then characterize Gorenstein-projective $R\mathcal{Q}$-modules in terms of the corresponding quiver $R$-representations; this generalizes a result obtained by Luo-Zhang to the case of not necessarily finitely generated $R\mathcal{Q}$-modules, and partially recover results due to Eno…
It is proved that for a commutative noetherian ring with dualizing complex the homotopy category of projective modules is equivalent, as a triangulated category, to the homotopy category of injective modules. Restricted to compact objects, this statement is a reinterpretation of Grothendieck's duality theorem. Using this equivalence it is proved that the (Verdier) quotient of the category of acyclic complexes of projectives by its subcategory of totally acyclic complexes and the corresponding category consisting of injective modules are equivalent. A new characterization is provided for comple…
A structure theorem of the group codes which are relative projective for the subgroup $\lbrace 1 \rbrace$ of $G$ is given. With this, we show that all such relative projective group codes in a fixed group algebra $RG$ are in bijection to the chains of projective group codes of length $\ell$ in the group algebra $\mathbb{F}G$, where $\mathbb{F}$ is the residue field of $R$. We use a given chain to construct the dual code in $RG$ and also derive the minimum Hamming weight as well as a lower bound of the minimum euclidean weight.
