Endomorphism Ring Research Articles

Bricks over preprojective algebras and join-irreducible elements in Coxeter groups

A (semi)brick over an algebra A is a module S such that its endomorphism ring End A ( S ) is a (product of) division algebra . For each Dynkin diagram Δ , there is a bijection from the Coxeter group W of type Δ to the set of semibricks over the preprojective algebra Π of type Δ , which is restricted to a bijection from the set of join-irreducible elements of W to the set of bricks over Π . This paper is devoted to giving an explicit description of these bijections in the case Δ = A n or D n . First, for each join-irreducible element w ∈ W , we describe the corresponding brick S ( w ) in terms of “Young diagram-like” notation. Next, we determine the canonical join representation w = ⋁ i = 1 m w i of an arbitrary element w ∈ W based on Reading's work, and prove that ⨁ i = 1 m S ( w i ) is the semibrick corresponding to w .

Topological models for stable motivic invariants of regular number rings

AbstractFor an infinity of number rings we express stable motivic invariants in terms of topological data determined by the complex numbers, the real numbers and finite fields. We use this to extend Morel’s identification of the endomorphism ring of the motivic sphere with the Grothendieck–Witt ring of quadratic forms to deeper base schemes.

Bi-clean and clean Hopf modules

<abstract><p>Let $ R $ be a commutative ring with multiplicative identity, $ C $ a coassociative and counital $ R $-coalgebra, $ B $ an $ R $-bialgebra. A clean comodule is a generalization and dualization of a clean module. An $ R $-module $ M $ is called a clean module if the endomorphism ring of $ M $ over $ R $ (denoted by $ End_{R}(M) $) is clean. Thus, any element of $ End_{R}(M) $ can be expressed as a sum of a unit and an idempotent element of $ End_{R}(M) $. Moreover, for a right $ C $-comodule $ M $, the endomorphism set of $ C $-comodule $ M $ denoted by $ End^{C}(M) $ is a subring of $ End_{R}(M) $. A $ C $-comodule $ M $ is a clean comodule if the $ End^{C}(M) $ is a clean ring. A Hopf module $ M $ over $ B $ is a $ B $-module and a $ B $-comodule that satisfies the compatible conditions. This paper considers the notions of a clean ring, clean module, clean coalgebra, and clean comodule in relation to the Hopf Module. We divide our discussion into two parts, i.e., clean and bi-clean Hopf modules. A $ B $-Hopf module $ M $ is said to be clean if the endomorphism ring of $ M $ is clean, and $ M $ is a bi-clean Hopf module if $ M $ is clean as a module over $ B $ and also clean as a comodule over $ B $. Moreover, we give sufficient conditions of (bi)-clean bialgebras and Hopf modules related to the cleanness concept of modules and comodules.</p></abstract>

On weakly α-shifting ring

For a ring endomorphism α, we introduce weakly α-shifting ring which is an extension of reduced as well as α-shifting ring. The notion of weakly α-shifting ring is a generalization of weak α-compatible ring. We investigate various properties of this ring including some kinds of examples in the process of development of this new concept.

WEAK and CO-WEAK BAER MODULES

الهدف من هذا البحث هو دراسة مفاهيم حلقات ومودولات بــــيـيـر الضعيفة وريكارت الضعيفة. لقد حصلنا على العديد من توصيفات حلقات ريكارت الضعيفة وقدمنا خصائصها. تمت دراسة العلاقة بين مودولات ريكارت الضعيفة (بيير الضعيفة) وحلقة الإندومورفيزمات الخاصة بها. لقد أثبتنا أن مودولات بيير الضعيفة التي تملك مجموعة غير منتهية من العناصر الجامدة المتعامدة غير الصفرية في حلقة الإندومورفيزمات الخاصة بها هي على وجه التحديد مودولات بيير. بالإضافة إلى ذلك، فإن حلقة الإندومورفيزمات الخاصة بمودولات ريكارت الضعيفة نصف الإسقاطية هي حلقة شبه جامدة وحلقة الإندومورفيزمات الخاصة بمودولات ريكارت المرافقة نصف الأفقية هي حلقة شبه جامدة. علاوة على ذلك، فقد بينا أن المودولات الحرة هي مودولات بيير الضعيفة إذا وفقط إذا كانت حلقة الإندومورفيزمات الخاصة بها هي حلقة بيير ضعيفة يسارية.

Polarizations of Abelian Varieties Over Finite Fields via Canonical Liftings

AbstractWe describe all polarizations for all abelian varieties over a finite field in a fixed isogeny class corresponding to a squarefree Weil polynomial, when one variety in the isogeny class admits a canonical lifting to characteristic zero, that is, a lifting for which the reduction morphism induces an isomorphism of endomorphism rings. Categorical equivalences between abelian varieties over finite fields and fractional ideals in étale algebras enable us to explicitly compute isomorphism classes of polarized abelian varieties satisfying some mild conditions. We also implement algorithms to perform these computations.

Realizing corners of Leavitt path algebras as Steinberg algebras, with corresponding connections to graph C⁎-algebras

We show that the endomorphism ring of any nonzero finitely generated projective module over the Leavitt path algebra LK(E) of an arbitrary graph E with coefficients in a field K is isomorphic to a Steinberg algebra. This yields in particular that every nonzero corner of the Leavitt path algebra of an arbitrary graph is isomorphic to a Steinberg algebra. This in its turn gives that every K-algebra with local units which is Morita equivalent to the Leavitt path algebra of a row-countable graph is isomorphic to a Steinberg algebra. Moreover, we prove that a corner by a projection of a C⁎-algebra of a countable graph is isomorphic to the C⁎-algebra of an ample groupoid.

On the Monoid of Unital Endomorphisms of a Boolean Ring

Let X be a nonempty set and P(X) the power set of X. The aim of this paper is to provide an explicit description of the monoid End1P(X)(P(X)) of unital ring endomorphisms of the Boolean ring P(X) and the automorphism group Aut(P(X)) when X is finite. Among other facts, it is shown that if X has cardinality n≥1, then End1P(X)(P(X))≅Tnop, where Tn is the full transformation monoid on the set X and Aut(P(X))≅Sn.

Reprint of: 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.

“Infinite” Properties of Certain Local Cohomology Modules of Determinantal Rings

For given integers $m,n \geq 2$ there are examples of ideals $I$ of complete determinantal local rings $(R,\mathfrak{m}), \dim R = m+n-1, \operatorname{grade} I = n-1,$ with the canonical module $\omega_R$ and the property that the socle dimensions of $H^{m+n-2}_I(\omega_R)$ and $H^m_{\mathfrak{m}}(H^{n-1}_I(\omega_R))$ are not finite. In the case of $m = n$, i.e. a Gorenstein ring, the socle dimensions provide further information about the $\tau$-numbers as studied in \cite{MS}. Moreover, the endomorphism ring of $H^{n-1}_I(\omega_R)$ is studied and shown to be an $R$-algebra of finite type but not finitely generated as $R$-module generalizing an example of \cite{Sp6}.

Orders and polytropes: matrix algebras from valuations

We apply tropical geometry to study matrix algebras over a field with valuation. Using the shapes of min-max convexity, known as polytropes, we revisit the graduated orders introduced by Plesken and Zassenhaus. These are classified by the polytrope region. We advance the ideal theory of graduated orders by introducing their ideal class polytropes. This article emphasizes examples and computations. It offers first steps in the geometric combinatorics of endomorphism rings of configurations in affine buildings.

Abelian p-groups with minimal full inertia

The class of abelian p-groups with minimal full inertia, that is, satisfying the property that fully inert subgroups are commensurable with fully invariant subgroups is investigated, as well as the class of groups not satisfying this property; it is known that both the class of direct sums of cyclic groups and that of torsion-complete groups are of the first type. It is proved that groups with “small" endomorphism ring do not satisfy the property and concrete examples of them are provided via Corner’s realization theorems. Closure properties with respect to direct sums of the two classes of groups are also studied. A topological condition of the socle and a structural condition of the Jacobson radical of the endomorphism ring of a p-group G, both of which are satisfied by direct sums of cyclic groups and by torsion-complete groups, are shown to be independent of the property of having minimal full inertia. The new examples of fully inert subgroups, which are proved not to be commensurable with fully invariant subgroups, are shown not to be uniformly fully inert.

ABELIAN GROUPS WITH LEFT COMORPHIC ENDOMORPHISM RINGS

A ring $R$ is called left comorphic if for every $a\in R$ there exists $b\in
 R$ such that the left and right annihilators satisfy $Ra=l(b)$ and
 $r(a)=bR$. In this paper, the Abelian groups with left comorphic
 endomorphism rings are completely determined.

Small-Essentially Pseudo-Injective Modules

Let R be associative ring with unit element and X be unitary right R-module. In this work, we introduce the definition of the concept small-essentially pseudo injective module (shortly, S-Ess-pseudo injective). Many properties of this concept are introduced and also we are consider some of their characterizations. Furthermore, we are studied the relation between our concept and some known R-modules and give some results on their endomorphism rings.

Identifying central endomorphisms of an abelian variety via Frobenius endomorphisms

Assuming the Mumford–Tate conjecture, we show that the center of the endomorphism ring of an abelian variety defined over a number field can be recovered from an appropriate intersection of the fields obtained from its Frobenius endomorphisms. We then apply this result to exhibit a practical algorithm to compute this center.

On Semiregular Endomorphism Rings and the Dual of Yamagata’s Theorem

On Semiregular Endomorphism Rings and the Dual of Yamagata’s Theorem

Finding endomorphisms of Drinfeld modules

We give an effective algorithm to determine the endomorphism ring of a Drinfeld module, both over its field of definition and over a separable or algebraic closure thereof. Using previous results we deduce an effective description of the image of the adelic Galois representation associated to the Drinfeld module, up to commensurability. We also give an effective algorithm to decide whether two Drinfeld modules are isogenous, again both over their field of definition and over a separable or algebraic closure thereof. Questions of efficiency are left completely out of consideration.

Endomorphism Rings Via Minimal Morphisms

We prove that if \(u:K \rightarrow M\) is a left minimal extension, then there exists an isomorphism between two subrings, \({\text {End}}_R^M(K)\) and \({\text {End}}_R^K(M)\) of \({\text {End}}_R(K)\) and \({\text {End}}_R(M)\), respectively, modulo their Jacobson radicals. This isomorphism is used to deduce properties of the endomorphism ring of K from those of the endomorphism ring of M in certain situations such us when K is invariant under endomorphisms of M, or when K is invariant under automorphisms of M.

The growth of the discriminant of the endomorphism ring of the reduction of a rank 2 generic Drinfeld module

For q an odd prime power, A=Fq[T], and F=Fq(T), let ψ:A→F{τ} be a Drinfeld A-module over F of rank 2 and without complex multiplication, and let p=pA be a prime of A of good reduction for ψ, with residue field Fp. We study the growth of the absolute value |Δp| of the discriminant of the Fp-endomorphism ring of the reduction of ψ modulo p and prove that, for all p, |Δp| grows with |p|. Moreover, we prove that, for a density 1 of primes p, |Δp| is as close as possible to its upper bound |ap2−4μpp|, where X2+apX+μpp∈A[X] is the characteristic polynomial of τdegp.

On the computation of the endomorphism rings of abelian surfaces

We extend the result of Bisson [4] to compute the endomorphism rings of principally polarized, absolutely simple, and ordinary abelian surfaces defined over finite fields in subexponential time in the size of the base field. The abelian surfaces covered here were excluded in [4]. This is accomplished by using techniques introduced in [17] to efficiently determine overorders for Bass and Gorenstein orders. In addition we show that the endomorphism rings of certain principally polarized, absolutely simple, and ordinary abelian surfaces are not computable in subexponential time with current methods [4,21], including ours.