Arbitrary Ring Research Articles

Overgroups of elementary block-diagonal subgroups in the classical symplectic group over an arbitrary commutative ring

In this paper we prove a sandwich classification theorem for subgroups of the classical symplectic group over an arbitrary commutative ring $R$ that contain the elementary block-diagonal (or subsystem) subgroup $\operatorname{Ep}(\nu, R)$ corresponding to a unitary equivalence realation $\nu$ such that all self-conjugate equivalence classes of $\nu$ are of size at least 4 and all not-self-conjugate classes of $\nu$ are of size at least 5. Namely, given a subgroup $H$ of $\operatorname{Sp}(2n, R)$ such that $\operatorname{Ep}(\nu, R) \le H$ we show that there exists a unique exact major form net of ideals $(\sigma, \Gamma)$ over $R$ such that $\operatorname{Ep}(\sigma, \Gamma) \le H \le N_{\operatorname{Sp}(2n,R)}(\operatorname{Sp}(\sigma, \Gamma))$. Further, we describe the normalizer $N_{\operatorname{Sp}(2n,R)}(\operatorname{Sp}(\sigma, \Gamma))$ in terms of congruences.

Dg analogues of the Zuckerman functors and the dual Zuckerman functors I

We study the category of dg Harish-Chandra modules over arbitrary commutative rings, and generalize the induction functor, the production functor, the Zuckerman functor and the dual Zuckerman functor.

The existence and representation of the Drazin inverse of a 2 × 2 block matrix over a ring

In this paper, further results on the Drazin inverse are obtained in a ring. Several representations of the Drazin inverse of [Formula: see text] block matrices over an arbitrary ring are given under new conditions. Also, upper bounds for the Drazin index of block matrices are studied. Numerical examples are given to illustrate our results. Necessary and sufficient conditions for the existence as well as the expression of the group inverse of block matrices are obtained under certain conditions. In particular, some results of related papers which were considered for complex matrices, operator matrices and matrices over a skew field are extended to more general setting.

A Jordan canonical form for nilpotent elements in an arbitrary ring

In this paper we give an inductive new proof of the Jordan canonical form of a nilpotent element in an arbitrary ring. If a∈R is a nilpotent element of index n with von Neumann regular an−1, we decompose a=ea+(1−e)a with ea∈eRe≅Mn(S) a Jordan block of size n over a corner S of R, and (1−e)a nilpotent of index <n for an idempotent e of R commuting with a. This result makes it possible to characterize prime rings of bounded index n with a nilpotent element a∈R of index n and von Neumann regular an−1 as a matrix ring over a unital domain.

Endomorphisms of the poset of idempotent matrices

Let D be a division ring, n≥3 an integer, and Pn(D) the poset of all n×n idempotent matrices over D with the partial order defined by P≤Q if PQ=QP=P. Let T∈Mn(D) be an invertible matrix and σ:D→D an endomorphism (τ:D→D an anti-endomorphism). For any P∈Pn(D) we denote by Pσ (Pτ) the idempotent matrix obtained from P by applying σ (τ) entrywise. The map ϕ:Pn(D)→Pn(D) defined by ϕ(P)=TPσT−1, P∈Pn(D) (ϕ(P)=Tt(Pτ)T−1, P∈Pn(D)) is an injective map preserving order in both directions. Every such map is called a standard map. It has been known before that if D is an EAS division ring, then every injective order preserving map ϕ:Pn(D)→Pn(D) is either standard or of a very special degenerate form. In this paper we use some ideas from geometry of algebraic homogeneous spaces and elementary field theory to give examples showing that the EAS assumption is indispensable. Then we define generalized standard maps and using them we describe the general form of injective order preservers on Pn(D) for an arbitrary division ring D. Our proof is shorter than the original one for the special case of EAS division rings. Under somewhat stronger assumptions we get two characterizations of standard maps. All the results are optimal as shown by counterexamples.

The set of strong torsion elements of a module over a noncommutative ring

Let [Formula: see text] be an arbitrary ring and [Formula: see text] be a nonzero right [Formula: see text]-module. In this paper, we introduce the set [Formula: see text], for some nonzero ideal [Formula: see text] of [Formula: see text] of strong torsion elements of [Formula: see text] and the properties of this set are investigated. In particular, we are interested when [Formula: see text] is a submodule of [Formula: see text] and when it is a union of prime submodules of [Formula: see text].

On a partially ordered set associated to ring morphisms

We associate to any ring R with identity a partially ordered set Hom(R), whose elements are all pairs (a,M), where a=ker⁡φ and M=φ−1(U(S)) for some ring morphism φ of R into an arbitrary ring S. Here U(S) denotes the group of units of S. The assignment R↦Hom(R) turns out to be a contravariant functor of the category Ring of associative rings with identity to the category ParOrd of partially ordered sets. The maximal elements of Hom(R) constitute a subset Max(R) which, for commutative rings R, can be identified with the Zariski spectrum Spec(R) of R. Every pair (a,M) in Hom(R) has a canonical representative, that is, there is a universal ring morphism ψ:R→S(R/a,M/a) corresponding to the pair (a,M), where the ring S(R/a,M/a) is constructed as a universal inverting R/a-ring in the sense of Cohn. Several properties of the sets Hom(R) and Max(R) are studied.

Annihilators of polynomials

In 1942, McCoy proved that if a polynomial f(x) over a commutative ring R has a nonzero annihilator in then it has a nonzero annihilator in R also. Later in 1957 McCoy proved that for R an arbitrary ring, if is annihilated on the right by some nonzero polynomial in then is also annihilated on the right by some nonzero element of R. Generalizing these results of McCoy we prove that a polynomial f(x) over an arbitrary ring R has a nonzero right annihilator in R if and only if there exists a nonzero polynomial g(x) in such that for every c in the multiplicative monoid generated by the coefficients of f(x). We also show that taking c to be identity and coefficients of f(x) does not suffice. We also give a new proof of Nielsen’s result that every reversible ring is right McCoy. We prove that an exchange right linearly McCoy ring is reduced modulo the Jacobson radical. We also prove that is a McCoy ring, where S and T are local rings with nonzero Jacobson radicals in which product of two non-units is zero and is a bimodule such that and MJ(T) = 0. This leads to a large class of exchange non-abelian McCoy rings.

Essential-small submodules relative to an arbitrary submodule

Let R be an arbitrary ring and T a submodule of an R-module M. A submodule N is said to be Te-small in M, if for each essential submodule X of M, T ⊆ N + X implies that T ⊆ X. In this work we study this mentioned notion which is a generalization of the essential-small submodules as well as the T-small submodule. We use this notion to investigate T-essential radical of module, also to introduce generalized T-essential Hopfian modules.

Almost prime and weakly prime submodules

Let [Formula: see text] be an arbitrary ring and [Formula: see text] be a right [Formula: see text]-module. A proper submodule [Formula: see text] of [Formula: see text] is called almost prime (respectively, weakly prime) if for each submodule [Formula: see text] of [Formula: see text] and each ideal [Formula: see text] of [Formula: see text] that [Formula: see text] and [Formula: see text] (respectively, [Formula: see text]), then [Formula: see text] or [Formula: see text]. We study these notions which are new generalizations of the prime submodules over noncommutative rings and we obtain some related results. We show that these two concepts in some classes of modules coincide. Moreover, we investigate the conditions that [Formula: see text] is almost prime, where [Formula: see text] is a submodule of [Formula: see text] and [Formula: see text] is an ideal of [Formula: see text]. Also, the almost prime radical of modules will be introduced and we extend some known results.

Commutativity in the lattice of topologizing filters of a commutative semiartinian ring

The set of all right topologizing filters on a fixed but arbitrary ring R admits a monoid operation ‘’ that is in general noncommutative, even in cases where the ring R is commutative. Earlier results show (see [8]) that commutativity of the monoid operation ‘’, when imposed as a condition on manifests as a type of finiteness condition on R. In a quite separate and much earlier study, Shores [6] has shown that if R is a commutative semiartinian ring, then R will be artinian precisely if the first two terms in the Loewy series for RR, namely and are finitely generated. Shores goes further to produce examples which show that the finiteness of just exercises no constraint whatsoever on the length of RR. The main result of this paper asserts that a commutative semiartinian ring R will be artinian precisely if is finitely generated and the monoid operation ‘’ on is commutative. A family of commutative semiartinian rings of Loewy length 3 is constructed and this used to delineate earlier theory. In particular, and within this family, rings R are exhibited such that (1) and have infinite length, yet (2) the monoid operation ‘’ on is commutative.

Compactly generated t-structures in the derived category of a commutative ring

We classify all compactly generated t-structures in the unbounded derived category of an arbitrary commutative ring, generalizing the result of Alonso Tarrio et al. (J Algebra 324(3):313–346, 2010) for noetherian rings. More specifically, we establish a bijective correspondence between the compactly generated t-structures and infinite filtrations of the Zariski spectrum by Thomason subsets. Moreover, we show that in the case of a commutative noetherian ring, any bounded below homotopically smashing t-structure is compactly generated. As a consequence, all cosilting complexes are classified up to equivalence.

Homological dimension of elementary amenable groups

Abstract In this paper we prove that the homological dimension of an elementary amenable group over an arbitrary commutative coefficient ring is either infinite or equal to the Hirsch length of the group. Established theory gives simple group theoretical criteria for finiteness of homological dimension and so we can infer complete information about this invariant for elementary amenable groups. Stammbach proved the special case of solvable groups over coefficient fields of characteristic zero in an important paper dating from 1970.

Correspondence between trace ideals and birational extensions with application to the analysis of the Gorenstein property of rings

Over an arbitrary commutative ring, correspondences among three sets, the set of trace ideals, the set of stable ideals, and the set of birational extensions of the base ring, are studied. The correspondences are well-behaved, if the base ring is a Gorenstein ring of dimension one. It is shown that with one extremal exception, the surjectivity of one of the correspondences characterizes the Gorenstein property of the base ring, provided it is a Cohen-Macaulay local ring of dimension one. Over a commutative Noetherian ring, a characterization of modules in which every submodule is a trace module is given. The notion of anti-stable rings is introduced, exploring their basic properties.

Leavitt path algebras over arbitrary unital rings and algebras

We expand the work of Tomforde by further extending the construction of Leavitt path algebras (LPAs) over arbitrary associative, unital rings. We show that many of the results over a commutative ring hold in the more general setting, provide some useful generalizations of prior results, and give a definition for an iterated Leavitt path extension in our context.

Dimension on non-essential submodules

In this paper, we introduce and study the concepts of non-essential Krull dimension and non-essential Noetherian dimension of an [Formula: see text]-module, where [Formula: see text] is an arbitrary associative ring. These dimensions are ordinal numbers and extend the notion of Krull dimension. They respectively rely on the behavior of descending and ascending chains of non-essential submodules. It is proved that each module with non-essential Krull dimension (respectively, non-essential Noetherian dimension) has finite Goldie dimension. We also show that a semiprime ring [Formula: see text] with non-essential Noetherian dimension is uniform.

Injective stabilization of additive functors, I. Preliminaries

This paper is the first one in a series of three dealing with the concept of injective stabilization of the tensor product and its applications. Its primary goal is to collect known facts and establish a basic operational calculus that will be used in the subsequent parts. This is done in greater generality than is necessary for the stated goal. Several results of independent interest are also established. They include, among other things, connections with satellites, an explicit construction of the stabilization of a finitely presented functor, various exactness properties of the injectively stable functors, a construction, from a functor and a short exact sequence, of a doubly-infinite exact sequence by splicing the injective stabilization of the functor and its derived functors. When specialized to the tensor product with a finitely presented module, the injective stabilization with coefficients in the ring is isomorphic to the 1-torsion functor. The Auslander-Reiten formula is extended to a more general formula, which holds for arbitrary (i.e., not necessarily finite) modules over arbitrary associative rings with identity. Weakening of the assumptions in the theorems of Eilenberg and Watts leads to characterizations of the requisite zeroth derived functors.The subsequent papers, provide applications of the developed techniques. Part II deals with new notions of torsion module and cotorsion module of a module. This is done for arbitrary modules over arbitrary rings. Part III introduces a new concept, called the asymptotic stabilization of the tensor product. The result is closely related to different variants of stable homology (these are generalizations of Tate homology to arbitrary rings). A comparison transformation from Vogel homology to the asymptotic stabilization of the tensor product is constructed and shown to be epic.

Dimension of Non-small Submodules

In this article, we introduce and study the concepts of non-small Krull dimension and non-small Noetherian dimension of an R-module M, where R is an arbitrary associative ring. These dimensions are ordinal numbers and extend the notion of Krull dimension of modules. They, respectively, rely on the behavior of descending and ascending chains of non-small submodules. We show that, if the non-small Noetherian dimension (resp., non-small Krull dimension) of an R-module M equals $$\alpha $$ , then $$\frac{M}{\mathrm{Rad}(M)}$$ has Noetherian dimension (resp., Krull dimension) and its Noetherian dimension (resp., Krull dimension) is less than or equal to $$\alpha $$ (resp., $$ \alpha +1$$ ). For decomposable modules, we show that the non-small Krull dimension (resp., non-small Noetherian dimension) and the Krull dimension (resp., Noetherian dimension) coincide.

Almost Bézout rings and almost GCD-rings

In this paper, we study the possible transfer of the property of being an [Formula: see text]-ring to trivial ring extensions and amalgamated algebras along an ideal. Also, we extend the notion of an almost GCD-domain to the context of arbitrary rings, and we study the possible transfer of this notion to trivial ring extensions and amalgamated algebras along an ideal. Our aim is to provide examples of new classes of commutative rings satisfying the above-mentioned properties.

The sums of exceptional units in a finite ring

We prove a formula for the number of representations of an element in a finite basic ring as a sum of k exceptional units and find bounds for this number in an arbitrary finite ring with identity.