Endomorphism Ring Research Articles

Centrally essential endomorphism rings of abelian groups

We study Abelian groups A with centrally essential endomorphism ring If A is a such group which is either a torsion group or a non-reduced group, then the ring is commutative. We give examples of Abelian torsion-free groups of finite rank with non-commutative centrally essential endomorphism rings.

Abelian endoregular modules

In this paper, we introduce the notion of abelian endoregular modules as those modules whose endomorphism rings are abelian von Neumann regular. We characterize an abelian endoregular module [Formula: see text] in terms of its [Formula: see text]-generated submodules. We prove that if [Formula: see text] is an abelian endoregular module then so is every [Formula: see text]-generated submodule of [Formula: see text]. Also, the case when the (quasi-)injective hull of a module as well as the case when a direct sum of modules is abelian endoregular are presented. At the end, we study abelian endoregular modules as subdirect products of simple modules.

Polynomial extensions of modules with the quasi-Baer property

In this paper it is shown that, for a module M over a ring R with S=EndR(M), the endomorphism ring of the R[x]-module M[x] is isomorphic to a subring of S[[x]]. Also the endomorphism ring of the R[[x]]-module M[[x]] is isomorphic to S[[x]]. As a consequence, we show that for a module MR and an arbitrary nonempty set of not necessarily commuting indeterminates X, MR is quasi-Baer if and only if M[X]R[X] is quasi-Baer if and only if M[[X]]R[[X]] is quasi-Baer if and only if M[x]R[x] is quasi-Baer if and only if M[[x]]R[[x]] is quasi-Baer. Moreover, a module MR with IFP, is Baer if and only if M[x]R[x] is Baer if and only if M[[x]]R[[x]] is Baer. It is also shown that, when MR is a finitely generated module, and every semicentral idempotent in S is central, then M[[X]]R[[X]] is endo-p.q.-Baer if and only if M[[x]]R[[x]] is endo-p.q.-Baer if and only if MR is endo-p.q.-Baer and every countable family of fully invariant direct summand of M has a generalized countable join. Our results extend several existing results.

Modules and their endomorphism rings

Let R and S be a pair of equivalent rings. We will prove that if R has property that a pair of modules over R are isomorphic if and only if their endomorphism rings are isomorphic than S has same property, i.e. a pair of modules over S are isomorphic if and only if their endomorphism rings are isomorphic.

Around Zero-Divisor Graph of Skew Polynomial Rings over Real Matrix 2 by 2

Let R be an associative ring with non-zero two-sided multiplicative identity. The zero-divisor of R, denoted by Z (R). The directed graph Γ(R) is a graph with vertices Z (R) − {0}, where x → y is an edge between distinct vertices x and y if and only if xy = 0. On the other hand, assume that σ is a ring endomorphism on R. The skew polynomial ring R[x; σ,] is the ring of polynomials (with indeterminate x) over R. In this paper, in the case that R = M2(ℝ), we study the zero-divisor graph of the skew polynomial ring R[x;σ].

Universal deformation rings of finitely generated Gorenstein-projective modules over finite dimensional algebras

Let k be a field of arbitrary characteristic, let Λ be a finite dimensional k-algebra, and let V be a finitely generated Λ-module. F. M. Bleher and the third author previously proved that V has a well-defined versal deformation ring R(Λ,V). If the stable endomorphism ring of V is isomorphic to k, they also proved under the additional assumption that Λ is self-injective that R(Λ,V) is universal. In this paper, we prove instead that if Λ is arbitrary but V is Gorenstein-projective then R(Λ,V) is also universal when the stable endomorphism ring of V is isomorphic to k. Moreover, we show that singular equivalences of Morita type (as introduced by X. W. Chen and L. G. Sun) preserve the isomorphism classes of versal deformation rings of finitely generated Gorenstein-projective modules over Gorenstein algebras. We also provide examples. In particular, if Λ is a monomial algebra in which there is no overlap (as introduced by X. W. Chen, D. Shen and G. Zhou) we prove that every finitely generated indecomposable Gorenstein-projective Λ-module has a universal deformation ring that is isomorphic to either k or to k〚t〛/(t2).

On commutator Krylov transitive and commutator weakly transitive Abelian p-groups

Abstract We define the concepts of commutator (Krylov) transitive and strongly commutator (Krylov) transitive Abelian p-groups. These two innovations are respectively non-trivial generalizations of the notions of commutator fully transitive and strongly commutator fully transitive p-groups from a paper of Chekhlov and Danchev (J. Group Theory, 2015). They are also commutator socle-regular in the sense of Danchev and Goldsmith (J. Group Theory, 2014). Various results from there and from a paper of Goldsmith and Strüngmann (Houston J. Math., 2007) are considerably extended to this new point of view. We also define and explore the concept of a commutator weakly transitive Abelian p-group, comparing its properties with those of the aforementioned two group classes. Some affirmations, sounding quite curiously, are detected in order to illustrate the pathology of the commutators in the endomorphism rings of p-primary Abelian groups.

Isogeny graphs with maximal real multiplication

An isogeny graph is a graph whose vertices are principally polarizable abelian varieties and whose edges are isogenies between these varieties. In his thesis, Kohel describes the structure of isogeny graphs for elliptic curves and shows that one may compute the endomorphism ring of an elliptic curve defined over a finite field by using a depth-first search (DFS) algorithm in the graph. In dimension 2, the structure of isogeny graphs is less understood and existing algorithms for computing endomorphism rings are very expensive. In this article, we show that, under certain conditions, the problem of determining the endomorphism ring can also be solved in genus 2 with a DFS-based algorithm. We consider the case of genus-2 Jacobians with complex multiplication, with the assumptions that the real multiplication subring has class number one and is locally maximal at ℓ, for ℓ a fixed prime. We describe the isogeny graphs in that case, by considering cyclic isogenies of degree ℓ, under the assumption that there is an ideal l of norm ℓ in K0 which is generated by a totally positive algebraic integer. The resulting algorithm is implemented over finite fields, and examples are provided. To the best of our knowledge, this is the first DFS-based algorithm in genus 2.

A note on the endomorphism ring of orthogonal modules

In this paper, we extend Mohamed−Mu¨ller’s results [2, Lemma 3.3] about the endomorphism ring of a module M = ⊕i2IMi; where Mi and Mj are orthogonal for all distinct elements i; j 2 I.

Isomorphism between endomorphism rings of modules over a semi simple ring

Our question is what ring R which all modules over R are determined, up to isomorphism, by their endomorphism rings? Examples of this ring are division ring and simple Artinian ring. Any semi simple ring does not satisfy this property. We construct a semi simple ring R but R is not a simple Artinian ring which all modules over R are determined, up to isomorphism, by their endomorphism rings.

Generalization of a connectedness result to cohomologically complete intersections

It is a well-known result from Hartshorne that, in projective space over a field, every set-theoretical complete intersection of positive dimension is connected in codimension one. Another important connectedness result (also from Hartshorne) is that a local ring with disconnected punctured spectrum has depth at most 1 1 . The two results are related; Hartshorne calls the latter “the keystone to the proof” of the former. In this short note we show how the latter result generalizes smoothly from set-theoretical to cohomologically complete intersections, i.e., to ideals for which there is in terms of local cohomology no obstruction to be a complete intersection. The proof is based on the fact that for cohomologically complete intersections over a complete local ring, the endomorphism ring of the (only) local cohomology module is the ring itself and hence is indecomposable as a module.

The cb-norm approximation of generalized skew derivations by elementary operators

ABSTRACT Let A be a ring and a ring endomorphism. A generalized skew (or σ-)derivation of A is an additive map for which there exists a map such that for all . If A is a prime -algebra and σ is surjective, we determine the structure of generalized σ-derivations of A that belong to the cb-norm closure of elementary operators on A; all such maps are of the form for suitable elements a,b,c of the multiplier algebra . As a consequence, if an epimorphism lies in the cb-norm closure of , then σ must be an inner automorphism. We also show that these results cannot be extended even to relatively well-behaved non-prime -algebras like .

Non-commutative resolutions of toric varieties

Let R be the coordinate ring of an affine toric variety. We prove, using direct elementary methods, that the endomorphism ring EndR(A), where A is the (finite) direct sum of all (isomorphism classes of) conic R-modules, has finite global dimension equal to the dimension of R. This gives a precise version, and an elementary proof, of a theorem of Špenko and Van den Bergh implying that EndR(A) has finite global dimension. Furthermore, we show that EndR(A) is a non-commutative crepant resolution if and only if the toric variety is simplicial. For toric varieties over a perfect field k of prime characteristic, we show that the ring of differential operators Dk(R) has finite global dimension.

Modules Whose Endomorphism Rings Are (m, n)-Coherent

Let M be a right R-module with endomorphism ring S. We study the left (m, n)-coherence of S. It is shown that S is a left (m, n)-coherent ring if and only if the left annihilator [Formula: see text] is a finitely generated left ideal of Mn(S) for any M-m-generated submodule X of Mn if and only if every M-(n, m)-presented right R-module has an add M-preenvelope. As a consequence, we investigate when the endomorphism ring S is left coherent, left pseudo-coherent, left semihereditary or von Neumann regular.

Universal deformation rings of string modules over certain class of self-injective special biserial algebras

We determine the universal deformation rings of string -modules whose corresponding stable endomorphism ring is isomorphic to and which lie either in a connected component of the stable Auslander-Reiten quiver of containing a module with endomorphism ring isomorphic to or in a periodic component containing only string -modules, where and are integers, and is a self-injective special biserial -algebra.

Algebras of generalized quaternion type

We introduce and study the algebras of generalized quaternion type, being natural generalizations of algebras which occurred in the study of blocks of group algebras with generalized quaternion defect groups. We prove that all these algebras, with 2-regular Gabriel quivers, are periodic algebras of period 4 and very specific deformations of the weighted surface algebras of triangulated surfaces with arbitrarily oriented triangles. The main result of the paper forms an important step towards the Morita equivalence classification of all periodic symmetric tame algebras of non-polynomial growth. Applying the main result, we establish existence of wild periodic algebras of period 4, with arbitrary large number (at least 4) of pairwise non-isomorphic simple modules. These wild periodic algebras arise as stable endomorphism rings of cluster tilting Cohen-Macaulay modules over one-dimensional hypersurface singularities.

Relations between endomorphism rings, injectivity, surjectivity and uniserial modules

For a right module [Formula: see text] over a ring [Formula: see text], we consider the set [Formula: see text] of all the endomorphisms [Formula: see text] that are not injective and the set [Formula: see text] of all the endomorphisms [Formula: see text] that are not surjective. We prove that when [Formula: see text] is a uniserial module, then [Formula: see text] is a left chain domain and [Formula: see text] is a right chain domain. The technique we make use of to prove this can be applied to arbitrary modules [Formula: see text], not-necessarily uniserial. When the endomorphisms [Formula: see text] are not in [Formula: see text] (not in [Formula: see text]), then left (right) divisibility in [Formula: see text] corresponds to inclusion in the lattice [Formula: see text] of all submodules of [Formula: see text]. This allows to study factorizations of injective (surjective, respectively) endomorphisms of [Formula: see text] analyzing finite chains in the partially ordered set [Formula: see text].

Principally quasi-Baer modules and their generalizations

The purpose of this article is to further the study of principally quasi-Baer modules and -principally quasi-Baer modules as these properties play an important role in the study of quasi-Baer module. First, we provide some basic results and a characterization of a principally quasi-Baer (simply, p.q.-Baer) module in terms of its endomorphism ring by using the pq-local-retractable property. In addition, we fully characterize when a finite direct sum of arbitrary p.q.-Baer modules is p.q.-Baer. Next, we obtain characterizations and properties of -principally quasi-Baer (simply, -p.q.-Baer) modules. Examples which show that the notion of an -p.q.-Baer module is distinct from that of a p.q.-Baer module are provided. It is shown that every direct summand of an -p.q.-Baer module inherits the property. Furthermore, we obtain that every direct sum of copies of an -p.q.-Baer module is an -p.q.-Baer module. We provide conditions when (-)p.q.-Baer modules become quasi-Baer modules. In particular, if every direct sum of copies of a module M is p.q.-Baer then the module M is a quasi-Baer module.

Endomorphism rings of reductions of Drinfeld modules

Let A=Fq[T] be the polynomial ring over Fq, and F be the field of fractions of A. Let ϕ be a Drinfeld A-module of rank r≥2 over F. For all but finitely many primes p◁A, one can reduce ϕ modulo p to obtain a Drinfeld A-module ϕ⊗Fp of rank r over Fp=A/p. The endomorphism ring Ep=EndFp(ϕ⊗Fp) is an order in an imaginary field extension K of F of degree r. Let Op be the integral closure of A in K, and let πp∈Ep be the Frobenius endomorphism of ϕ⊗Fp. Then we have the inclusion of orders A[πp]⊂Ep⊂Op in K. We prove that if EndFalg(ϕ)=A, then for arbitrary non-zero ideals n,m of A there are infinitely many p such that n divides the index χ(Ep/A[πp]) and m divides the index χ(Op/Ep). We show that the index χ(Ep/A[πp]) is related to a reciprocity law for the extensions of F arising from the division points of ϕ. In the rank r=2 case we describe an algorithm for computing the orders A[πp]⊂Ep⊂Op, and give some computational data.

Modules Whose Endomorphism Rings are Right Rickart

In this paper, we study modules whose endomorphism rings are right Rickart (or right p.p.) rings, which we call R-endoRickart modules. We provide some characterizations of R-endoRickart modules. Some classes of rings are characterized in terms of R-endoRickart modules. We prove that an R-endoRickart module with no innite set of nonzero orthogonal idempotents in its endomorphism ring is precisely an endoBaer module. We show that a direct summand of an R-endoRickart modules inherits the property, while a direct sum of R-endoRickart modules does not. Necessary and sucient conditions for a nite direct sum of R-endoRickart modules to be an R-endoRickart module are provided.