Let $R$ be an associative ring with identity. A unital right $R$-module $M$ is called strongly finite dimensional if Sup$\{{\rm G.dim} (M/N) | N\leq M\} < +\infty$. Properties of strongly finite dimensional modules are explored. It is also proved that: (1)If $R$ is left $F$-injective and strongly right finite dimensional, then $R$ is left finite dimensional. (2) If $R$ is right $F$-injective, then $R$ is right finite dimensional if and only if $R$ is semilocal. Thus the Faith-Menal conjecture is true if $R$ is strongly right finite dimensional. Some known results are obtained as corollaries.
We find a bound for the Goldie dimension of hereditary modules in terms of the cardinality of the generator sets of its quasi-injective hull. Several consequences are deduced. In particular, it is shown that every right hereditary module with countably generated quasi-injective hull is noetherian. Or that every right hereditary ring with finitely generated injective hull is artinian, thus answering a long standing open question posed by Dung, Gomez Pardo and Wisbauer.
Let $R$ be a ring, $σ$ an injective endomorphism of $R$ and $δ$ a $σ$-derivation of $R$. We prove that if $R$ is semiprime left Goldie then the same holds for the Ore extension $R[x;σ,δ]$ and both rings have the same left uniform dimension.
The lower transcendence degree, introduced by J. J Zhang, is an important non-commutative invariant in ring theory and non-commutative geometry strongly connected to the classical Gelfand-Kirillov transcendence degree. For LD-stable algebras, the lower transcendence degree coincides with the Gelfand-Kirillov dimension. We show that the following algebras are LD-stable and compute their lower transcendence degrees: rings of differential operators of affine domains, universal enveloping algebras of finite dimensional Lie superalgebras, symplectic reflection algebras and their spherical subalgebr…
We answer in negative two of questions posed in [4]. We also establish a new characterization of semiprime left Goldie rings by showing that a semiprime ring R is left Goldie iff it is regular left fusible and has finite left Goldie dimension.
It is known that any primitive ideal I of U(g) whose associated variety contains a nilpotent element e in its open G-orbit admits a finite generalised Gelfand-Graev model which is a finite dimensional irreducible module over the finite W-algebra U(g,e). We prove that if V is such a model for I, then the Goldie rank of the primitive quotient U(g)/I always divides the dimension of V. For g=sl(n), we use a result of Joseph to show that the Goldie rank of U(g)/I equals the dimension of V and we show that the equality conntinues to hold outside type A provided that the Goldie field of U(g)/I is iso…
Let $\mathfrak{g}$ be a semisimple Lie algebra. We establish a new relation between the Goldie rank of a primitive ideal $\mathcal{J}\subset U(\mathfrak{g})$ and the dimension of the corresponding irreducible representation $V$ of an appropriate finite W-algebra. Namely, we show that $\operatorname{Grk}(\mathcal{J}) \leqslant \dim V/d_V$, where $d_V$ is the index of a suitable equivariant Azumaya algebra on a homogeneous space. We also compute $d_V$ in representation theoretic terms.
It is proved that localizations of injective $R$-modules of finite Goldie dimension are injective if $R$ is an arithmetical ring satisfying the following condition: for every maximal ideal $P$, $R_P$ is either coherent or not semicoherent. If, in addition, each finitely generated $R$-module has finite Goldie dimension, then localizations of finitely injective $R$-modules are finitely injective too. Moreover, if $R$ is a Prüfer domain of finite character, localizations of injective $R$-modules are injective.
Every automorphism-invariant right non-singular $A$-module is injective if and only if the factor ring of the ring $A$ with respect to its right Goldie radical is a right strongly semiprime ring.
It is proved that localizations of injective $R$-modules of finite Goldie dimension are injective if $R$ is an arithmetical ring satisfying the following condition: for every maximal ideal $P$, $R_P$ is either coherent or not semicoherent. If, in addition, each finitely generated $R$-module has finite Goldie dimension, then localizations of finitely injective $R$-modules are finitely injective too. Moreover, if $R$ is a Prüfer domain of finite character, localizations of injective $R$-modules are injective.
$\textbf{Theorem 1.3.}$ For a given ring $A$ with right Goldie radical $G(A_A)$, the following conditions are equivalent. $\textbf{1)}$ Every non-singular right $A$-module $X$ which is is injective with respect to some essential right ideal of the ring $A$ is an injective module. $\textbf{2)}$ $A/G(A_A)$ is a right strongly semiprime ring. $\textbf{Theorem 1.4.}$ For a given ring $A$, the following conditions are equivalent. $\textbf{1)}$ $A$ is a right strongly semiprime ring. $\textbf{2)}$ Every right $A$-module which is injective with respect to some essential right ideal of the ring $A$, i…
$\textbf{Theorem 1.2.}$ For a ring $A$, the following conditions are equivalent. $\textbf{1)}$ $A$ is a right automorphism-invariant right non-singular ring. $\textbf{2)}$ $A$ is a right automorphism-invariant regular ring. $\textbf{3)}$ $A=S\times T$, where $S$ is a right injective regular ring and $T$ is a strongly regular ring which contains all invertible elements of its maximal right ring of quotients. $\textbf{Theorem 1.5.}$ For a ring $A$ with right Goldie radical $G(A_A)$, the following conditions are equivalent. $\textbf{1)}$ $A/G(A_A)$ is a semiprime right Goldie ring. $\textbf{2)}$ …
Let $k$ be an uncountable field. We show that a finitely generated prime Goldie $k$-algebra of quadratic growth is either primitive or satisfies a polynomial identity, answering a question of Small in the affirmative.
It is shown that a ring is left semihereditary if and only each homomorphic image of its injective hull as left module is FP-injective. It is also proven that a commutative ring R is reduced and arithmetical if and only if E/U if FP-injective for any FP-injective R-module E and for any submodule U of finite Goldie dimension. A characterization of commutative rings for which each module of finite Goldie dimension is of injective dimension at most one is given. Let R be a chain ring and Z its subset of zerodivisors. It is proven that E/U is FP-injective for each FP-injective R-module E and each …
The main aim of this paper is to show that an AB5*-module whose small submodules have Krull dimension has a radical having Krull dimension. The proof uses the notion of dual Goldie dimension.
Dimensions like Gelfand, Krull, Goldie have an intrinsic role in the study of theory of rings and modules. They provide useful technical tools for studying their structure. In this paper we define one of the dimensions called couniserial dimension that measures how close a ring or module is to being uniform. Despite their different objectives, it turns out that there are certain common properties between the couniserial dimension and Krull dimension like each module having such a dimension contains a uniform submodule and has finite uniform dimension, among others. Like all dimensions, this is…
Let $k$ be a field. We show that a finitely generated simple Goldie $k$-algebra of quadratic growth is noetherian and has Krull dimension 1. Thus a simple algebra of quadratic growth is left noetherian if and only if it is right noetherian. As a special case, we see that if A is a finitely generated simple domain of quadratic growth then A is noetherian and by a result of Stafford every right and left ideal is generated by at most two elements. We conclude by posing questions and giving examples in which we consider what happens when the hypotheses are relaxed.
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.
It is well-known that a ring R is semiperfect if and only if R as a left (or as a right) R-module is a supplemented module. Considering weak supplements instead of supplements we show that weakly supplemented modules M are semilocal (i.e., M/Rad(M) is semisimple) and that R is a semilocal ring if and only if R as a left (or as a right) R-module is weakly supplemented. In this context the notion of finite hollow dimension (or finite dual Goldie dimension) of modules is of interest and yields a natural interpretation of the Camps-Dicks characterization of semilocal rings. Finitely generated modu…
Many known results on finite von Neumann algebras are generalized, by purely algebraic proofs, to a certain class ${\mathcal C}$ of finite Baer *-rings. The results in this paper can also be viewed as a study of the properties of Baer *-rings in the class ${\mathcal C}$. First, we show that a finitely generated module over a ring from the class ${\mathcal C}$ splits as a direct sum of a finitely generated projective module and a certain torsion module. Then, we define the dimension of any module over a ring from ${\mathcal C}$ and prove that this dimension has all the nice properties of the di…
The dimension of any module over an algebra of affiliated operators ${\mathcal U}$ of a finite von Neumann algebra ${\mathcal A}$ is defined using a trace on ${\mathcal A}.$ All zero-dimensional ${\mathcal U}$-modules constitute the torsion class of torsion theory $(\mathrm{\bf T},\mathrm{\bf P})$. We show that every finitely generated ${\mathcal U}$-module splits as the direct sum of torsion and torsion-free part. Moreover, we prove that the theory $(\mathrm{\bf T},\mathrm{\bf P})$ coincides with the theory of bounded and unbounded modules and also with the Lambek and Goldie torsion theories.…
It is well known that a ring $R$ is right Kasch if each simple right $R$-module embeds in a projective right $R$-module. In this paper we study the dual notion and call a ring $R$ right dual Kasch if each simple right $R$-module is a homomorphic image of an injective right $R$-module. We prove that $R$ is right dual Kasch if and only if every finitely generated projective right $R$-module is coclosed in its injective hull. Typical examples of dual Kasch rings are self-injective rings, V-rings and commutative perfect rings. Skew group rings of dual Kasch rings by finite groups are dual Kasch if…
The study of modules over a finite von Neumann algebra ${\mathcal A}$ can be advanced by the use of torsion theories. In this work, some torsion theories for ${\mathcal A}$ are presented, compared and studied. In particular, we prove that the torsion theory $(\mathrm{\bf T},\mathrm{\bf P})$ (in which a module is torsion if it is zero-dimensional) is equal to both Lambek and Goldie torsion theories for ${\mathcal A}$. Using torsion theories, we describe the injective envelope of a finitely generated projective ${\mathcal A}$-module and the inverse of the isomorphism $K_0({\mathcal A})\to K_0({\…
The aim of this paper is to provide a criterion to determine, by quivers with relations, when an algebra has global dimension at most two. In order to do that, we introduce a new class of algebras of global dimension three, and we call them critical algebras. Furthermore we give a characterization of critical algebras by quivers with relations. Our main theorem states that if a strongly simply connected schurian algebra does not contain a critical algebra as a full subcategory, then it has global dimension at most two.
This paper is a continuation of the papers J. Pure Appl. Algebra, 210 (2007), 437--445 and J. Algebra Appl., 8 (2009), 219--227. Namely, we introduce and study a doubly filtered set of classes of modules of finite Gorenstein projective dimension, which are called $(n,m)$-strongly Gorenstein projective ($(n,m)$-SG-projective for short) for integers $n\geq 1$ and $m\geq 0$. We are mainly interested in studying syzygies of these modules. As consequences, we show that a module $M$ has Gorenstein projective dimension at most $m$ if and only if $M\oplus G$ is $(1,m)$-SG-projective for some Gorenstei…
