The centralizer of an element in an endomorphism ring

New Results on Generalization of Jordan centralizers over matrix rings

This paper presents a study on Jordan maps over matrix rings with some functional equations related to additive maps on these rings. We first show that every Jordan left (right) centralizer over a matrix ring is a left (right) centralizer. Moreover, every two-sided centralizer over the matrix ring is of a particular form. Further, we prove that any additive map satisfying functional equations over matrix rings becomes a two-sided centralizer. Finally, we conclude our work with some results on the Jordan left $\star$- centralizer over matrix rings and establish some results on functional equati…

非结合环中的中心化子与伪度数函数

Centralizers in non-associative rings with a pseudo-degree function

本文研究非结合代数中元素的中心化子，证明其为有限秩自由模。

Centralizers in Ore extensions of polynomial rings

In this paper we consider centralizers of single elements in Ore extensions of the ring of polynomials in one variable over a field. We show that they are commutative and finitely generated as an algebra. We also show that for certain classes of elements their centralizer is singly generated as an algebra.

Generalized (m,n)-Jordan centralizers and derivations on semiprime rings

In this article, we prove Conjecture $1$ posed in 2013 by Fo$\check{s}$ner \cite{fos} and Conjecture $1$ posed in 2014 by Ali and Fo$\check{s}$ner \cite{ali} related to generalized $(m,n)$-Jordan centralizer and derivation respectively.

On Generalized $(m, n)$-Jordan Derivations and Centralizers of Semiprime Rings

In this paper we give an affirmative answer to two conjectures on generalized $(m,n)$-Jordan derivations and generalized $(m,n)$-Jordan centralizers raised in [S. Ali and A. Fošner, \textit{On Generalized $(m, n)$-Derivations and Generalized $(m, n)$-Jordan Derivations in Rings,} Algebra Colloq. \textbf{21} (2014), 411--420] and [A. Fošner, \textit{A note on generalized $(m,n)$-Jordan centralizers,} Demonstratio Math. \textbf{46} (2013), 254--262]. Precisely, when $R$ is a semiprime ring, we prove, under some suitable torsion restrictions, that every nonzero generalized $(m,n)$-Jordan derivati…

Characterizing some rings of finite order

In this paper, we compute the number of distinct centralizers of some classes of finite rings. We then characterize all finite rings with $n$ distinct centralizers for any positive integer $n \leq 5$. Further we give some connections between the number of distinct centralizers of a finite ring and its commutativity degree.

Left centralizers on Lie ideals in prime and semiprime gamma rings

Let $U$ be a Lie ideal of a 2-torsion free prime $Γ$-ring $M$ such that $uαu\in U$ for all $u\in U$ and $α\inΓ$. If $T:M\rightarrow M$ is an additive mapping satifying the relation $T(uαu)=T(u)αu$ for all $u\in U$ and $α\inΓ$, then we prove that $T(uαv)=T(u)αv$ for all $u, v\in U$ and $α\inΓ$. Also this result is extended to semiprime $Γ$-rings.

Characterizing Jordan centralizers and Jordan generalized derivations on triangular rings through zero products

Let $\T$ be a $2$-torsion free triangular ring and let $φ:\T\rightarrow \T$ be an additive map. We prove that if $\A φ(\B)+φ(\B)\A=0$ whenever $\A,\B\in \T$ are such that $\A\B=\B\A=0$, then $φ$ is a centralizer. It is also shown that if $τ:\T\rightarrow \T$ is an additive map satisfying $\label{t2} X,Y\in \T, \quad XY=YX=0\Rightarrow X τ(Y)+δ(X)Y+Yδ(X)+τ(Y)X=0$, where $δ:\T\rightarrow \T $ is an additive map satisfies $X,Y\in \T, \quad XY=YX=0\Rightarrow X δ(Y)+δ(X)Y+Yδ(X)+δ(Y)X=0$, then $τ(\A)=d(\A)+\A τ(\textbf{1})$, where $d:\T\rightarrow \T$ is a derivation and $τ(\textbf{1})$ lies in the…

Endomorphism rings of modules over prime rings

Endomorphism rings of modules appear as the center of a ring, as the fix ring of ring with group action or as the subring of constants of a derivation. This note discusses the question whether certain *-prime modules (introduced by Bican et al.) have a prime endomorphism ring. Several conditions are presented that guarantee the primness of the endomorphism ring. The contours of a possible example of a *-prime module whose endomorphism ring is not prime are traced.

Centrally Essential Endomorphism Rings of Abelian Groups

We study Abelian groups $A$ with centrally essential endomorphism ring $\text{End}\,A$. If $A$ is a such group which is either a torsion group or a non-reduced group, then the ring $\text{End}\,A$ is commutative. We give examples of Abelian torsion-free groups of finite rank with non-commutative centrally essential endomorphism rings. The paper will appear in Communications in Algebra.

Centralizer's applications to the inverse along an element

In this paper, we first prove that the absorption law for one-sided inverses along an element holds, deriving the absorption law for the inverse along an element. We then apply this result to obtain the absorption law for the inverse along different elements. Also, the reverse order law and the existence criterion for the inverse along an element are given by centralizers in a ring. Finally, we characterize the Moore-Penrose inverse by one-sided invertibilities in a ring with involution.

Centralizers on Prime and Semiprime Gamma Rings

Let $M$ be a noncommutative 2-torsion free semiprime $Γ$-ring satisfying a certain assumption and let $S$ and $T$ be left centralizers on $M$. We prove the following results: \\(i) If $[S(x),T(x)]_{α}βS(x)+S(x)β[S(x),T(x)]_{α}$=$0$ holds for all $x\in M$ and $α,β\in Γ$, then $[S(x),T(x)]_{α}$=$0$. \\(ii) If $S\neq 0 (T\neq 0)$, then there exists $λ\in C$,(the extended centroid of $M$) such that $T$=$λαS(S=λαT)$ for all $α\in Γ$. \\(iii) Suppose that $[[S(x),T(x)]_{α},S(x)]_{β}$=$0$ holds for all $x\in M$ and $α,β\inΓ$. Then $[S(x),T(x)]_{α}$=$0$ for all $x\in M$ and $α\inΓ$. \\(iv) If $M$ is a…

Ring Endomorphisms with Large Images

The notion of ring endomorphisms having large images is introduced. Among others, injectivity and surjectivity of such endomorphisms are studied. It is proved, in particular, that an endomorphism S of a prime one-sided noetherian ring R is injective whenever the image S (R) contains an essential left ideal L of R. If additionally S(L) = L, then S is an automorphism of R. Examples showing that the assumptions imposed on R can not be weakened to R being a prime left Goldie ring are provided. Two open questions are formulated.

The endomorphism ring of an injective square-free module

Let $M_R$ be an injective right module over a ring $R$. The goal of this paper to prove that the endomorphism ring $S=(M_R)$ of $M$ is quasi-duo if and only if $M_R$ is square-free.

Endomorphism rings of Leavitt path algebras

We investigate conditions under which the endomorphism ring of the Leavitt path algebra $L_{K}(E)$ possesses various ring and module-theoretical properties such as being von Neumann regular, $π$-regular, strongly $π$-regular or self-injective. We also describe conditions under which $L_{K}(E)$ is continuous as well as automorphism invariant as a right $L_{K}(E)$-module.

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, $\textrm{End}_R^M(K)$ and $\textrm{End}_R^K(M)$ of $\textrm{End}_R(K)$ and $\textrm{End}_R(M)$ 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 $M$.

关于一类自同态环至多有四个极大理想的扩张范畴

On a category of extensions whose endomorphism rings have at most four maximal ideals

该研究描述了A_R和C_R为单列模的短正合序列的自同态环，并探究了这些短正合序列在直和方面的行为，涉及自同态环至多含四个极大理想的扩张范畴。

Rings close to periodic with applications to matrix, endomorphism and group rings

We examine those matrix rings whose entries lie in periodic rings equipped with some additional properties. Specifically, we prove that the famous Diesl's question whether or not $R$ being nil-clean implies that $\mathbb{M}_n(R)$ is nil-clean for all $n\geq 1$ is paralleling to the corresponding implication for (Abelian, local) periodic rings. Besides, we study when the endomorphism ring $\mathrm{E}(G)$ of an Abelian group $G$ is periodic. Concretely, we establish that $\mathrm{E}(G)$ is periodic exactly when $G$ is finite as well as we find a complete necessary and sufficient condition when t…

通过一对同态的素环可交换性准则

Commutativity criteria for prime rings with involution via pairs of endomorphisms

本文通过研究素环中涉及一对同态的中心值恒等式，提出新的可交换性准则并构造反例，验证了素性和involution性质的必要性。

Direct Products of Modules Whose Endomorphism Rings have at Most Two Maximal Ideals

Monogeny classes and epigeny classes have proved to be useful in the study of direct sums of uniserial modules and other classes of modules. In this paper, we show that they also turn out to be useful in the study of direct products.

Local morphisms and modules with a semilocal endomorphism ring

An associative ring with 1 is said to be semilocal provided it is semisimple artinian modulo its Jacobson radical, that is, modulo its Jacobson radical it is isomorphic to a finite product of matrices over division rings. Modules with a semilocal endomorphism ring inherit some properties of semisimple modules. For example, they can be decomposed into a finite direct sum of indecomposable submodules, such decomposition is not unique but there are only a finite number of different ones, they cancel from direct sums and they satisfy the n-th root uniqueness property. A ring homomorphism is said t…

Generating Endomorphism Rings of Infinite Direct Sums and Products of Modules

Let R be a ring, M a left R-module, I an infinite set, N either the direct sum or product of |I| copies of M, and E the endomorphism ring of N as a left R-module. In this note it is shown that E is not the union of a chain of |I| or fewer proper subrings, and also that given a generating set U for E as a ring, there exists a positive integer n such that every element of E is represented by a ring word of length at most n in elements of U.

关于一类模链的范畴，其自同态环至多有$2n$个极大理想

On a category of chains of modules whose endomorphism rings have at most $2n$ maximal ideals

本文研究一类带固定子模链的右$R$-模构成的加法范畴，描述其自同态环性质并分析该范畴对象的直和行为。

单模的自同态环与块分解

Endomorphism rings of simple modules and block decomposition

本文利用左右诺特半完全环不可分解性的特征刻画，通过Ext群研究单模，证明不可分解半完全环的单模基数及自同态环特征的一致性，并在拟Frobenius环情形下强化了结果。

Poisson Centralizer of the Trace

The Poisson centralizer of the trace element is determined in the coordinate ring of SL_n endowed with the Poisson structure obtained as the semiclassical limit of its quantized coordinate ring. It turns out that this maximal Poisson-commutative subalgebra coincides with the subalgebra of invariants with respect to the adjoint action.

1-Primitive Near-rings

We give a classification of $1$-primitive near-rings using sandwich centralizer near-rings

Strongly clean triangular matrix rings with endomorphisms

A ring $R$ is strongly clean provided that every element in $R$ is the sum of an idempotent and a unit that commutate. Let $T_n(R,σ)$ be the skew triangular matrix ring over a local ring $R$ where $σ$ is an endomorphism of $R$. We show that $T_2(R,σ)$ is strongly clean if and only if for any $a\in 1+J(R), b\in J(R)$, $l_a-r_{σ(b)}: R\to R$ is surjective. Further, $T_3(R,σ)$ is strongly clean if $l_{a}-r_{σ(b)}, l_{a}-r_{σ^2(b)}$ and $l_{b}-r_{σ(a)}$ are surjective for any $a\in U(R),b\in J(R)$. The necessary condition for $T_3(R,σ)$ to be strongly clean is also obtained.

Additive Unit Representations in Endomorphism Rings and an Extension of a result of Dickson and Fuller

A module is called automorphism-invariant if it is invariant under any automorphism of its injective hull. Dickson and Fuller have shown that if $R$ is a finite-dimensional algebra over a field $\mathbb F$ with more than two elements then an indecomposable automorphism-invariant right $R$-module must be quasi-injective. In this note, we extend and simplify the proof of this result by showing that any automorphism-invariant module over an algebra over a field with more than two elements is quasi-injective. Our proof is based on the study of the additive unit structure of endomorphism rings.

Extension of Almost Armendariz Rings

A ring $R$ is said to be an almost Armendariz ring if whenever product of two polynomials in $R[x]$ is zero, then product of their coefficients are in $N_{*}(R)$. In this article, for an endomorphism $α$ on $R$, we define an $α$-almost Armendariz ring of $R$ considering the polynomials in skew polynomial ring $R[x; α]$ instead of $R[x]$. It is the generalisation of an almost Armendariz ring [9] and an $α$-Armendariz ring [4]. Moreover, for an endomorphism $α$ of $R$, we define an $α$-skew almost Armendariz ring, and prove that a reversible ring $R$ with certain condition on endomorphism $α$, i…

The centralizer of an $I$-matrix in $M_2(R/I)$, $R$ a UFD

The concept of an $I$-matrix in the full $2\times 2$ matrix ring $M_2(R/I)$, where $R$ is an arbitrary UFD and $I$ is a nonzero ideal in $R$, is introduced. We obtain a concrete description of the centralizer of an $I$-matrix $\hat B$ in $M_2(R/I)$ as the sum of two subrings $\mathcal S_1$ and $\mathcal S_2$ of $M_2(R/I)$, where $\mathcal S_1$ is the image (under the natural epimorphism from $M_2(R)$ to $M_2(R/I)$) of the centralizer in $M_2(R)$ of a pre-image of $\hat B$, and where the entries in $\mathcal S_2$ are intersections of certain annihilators of elements arising from the entries of …

具有 Lie 理想的（斜）迹的非平凡中心化子的素环

Prime rings having nontrivial centralizers of (skew) traces of Lie ideals

本文研究了带对合的素环中非交换 Lie 理想的迹与斜迹在非零导子作用下为零的性质，证明了此时环的迹或斜迹的平方包含于中心，并将结论推广至除环和含幂等元的素环的特定子群，推广了近期相关成果。

Additive unit structure of endomorphism rings and invariance of modules

We use the type theory for rings of operators due to Kaplansky to describe the structure of modules that are invariant under automorphisms of their injective envelopes. Also, we highlight the importance of Boolean rings in the study of such modules. As a consequence of this approach, we are able to further the study initiated by Dickson and Fuller regarding when a module invariant under automorphisms of its injective envelope is invariant under any endomorphism of it. In particular, we find conditions for several classes of noetherian rings which ensure that modules invariant under automorphis…

Completely simple endomorphism rings of modules

It is proved that if A_p is a countable elementary abelian p-group, then: (i) The ring End(A_p) does not admit a nondiscrete locally compact ring topology. (ii) Under (CH) the simple ring End(A_p)/I, where I is the ideal of End(A_p) consisting of all endomorphisms with finite images, does not admit a nondiscrete locally compact ring topology. (iii) The finite topology on End(A_p) is the only second metrizable ring topology on it. Moreover, a characterization of completely simple endomorphism rings of the endomorphism rings of modules over commutative rings is also obtained.

模之间具有半局部自同态环的同态

Homomorphisms with semilocal endomorphism rings between modules

该研究探讨了范畴$\text{Morph}(\text{Mod} R)$中自同态环为半局部的对象的行为，发现其直和分解依赖于特定不变量，且与具有半局部自同态环的模及有限Goldie维数序列模的分解行为相似。

A note on the singularity category of an endomorphism ring

We propose the notion of partial resolution of a ring, which is by definition the endomorphism ring of a certain generator of the given ring. We prove that the singularity category of the partial resolution is a quotient of the singularity category of the given ring. Consequences and examples are given.

Endomorphism rings generated using small numbers of elements

Let R be a ring, M a nonzero left R-module, X an infinite set, and E the endomorphism ring of the direct sum of copies of M indexed by X. Given two subrings S and S' of E, we will say that S is equivalent to S' if there exists a finite subset U of E such that the subring generated by S and U is equal to the subring generated by S' and U. We show that if M is simple and X is countable, then the subrings of E that are closed in the function topology and contain the diagonal subring of E (consisting of elements that take each copy of M to itself) fall into exactly two equivalence classes, with re…

A variant of Baer's theorem

We provide a variant of Baer's theorem about isomorphism of endomorphism rings of vector spaces over division rings, where the full endomorphism rings are replaced by some subrings of finitary maps.

Elementary Equivalence of Categories of Modules over Rings, Endomorphism Rings, and Automorphism Groups of Modules

In this paper we give a small review of some recent results of elementary equivalence of linear and algebraic groups and our last new results of elementary equivalence of categories of modules, endomorphism rings of modules, lattices of submodules of modules, and automorphism groups of modules.

Automorphisms for skew PBW extensions and skew quantum polynomial rings

In this work we study the automorphisms of skew $PBW$ extensions and skew quantum polynomials. We use Artamonov's works as reference for getting the principal results about automorphisms for generic skew $PBW$ extensions and generic skew quantum polynomials. In general, if we have an endomorphism on a generic skew $PBW$ extension and there are some $x_i,x_j,x_u$ such that the endomorphism is not zero on this elements and the principal coefficients are invertible, then endomorphism act over $x_i$ as $a_ix_i$ for some $a_i$ in the ring of coefficients. Of course, this is valid for quantum polyno…

Centralizers in Free Group Algebras and Nonsingular Curves

The centralizer of any non-scalar element of a free group algebra over a field is the coordinate ring of a nonsingular curve.

Centralizers of linear and locally nilpotent derivations

Let $K$ be an algebraically closed field of characteristic zero, $A = K[x_1,\dots,x_n]$ the polynomial ring, $R = K(x_1,\dots,x_n)$ the field of rational functions, and let $W_n(K) = \Der_{K}A$ be the Lie algebra of all $K$-derivations on $A$. If $D \in W_n(K),$ $D\not =0$ is linear (i.e. of the form $D = \sum_{i,j=1}^n a_{ij}x_j \frac{\partial}{\partial x_i}$) we give a description of the centralizer of $D$ in $W_n(K)$ and point out an algorithm for finding generators of $C_{W_n(K)}(D)$ as a module over the ring of constants in case when $D$ is the basic Weitzenboeck derivation. In more gener…

Commutators in the Endomorphism Ring of a Complex

According to an old result of Albert and Muckenhoupt, the commutators in the endomorphism ring of a finite dimensional vector space are precisely the elements of trace zero. We replace the finite dimensional vector space with a complex of finite dimensional vector spaces, and characterize commutators and other elements with commutator-like properties in terms of appropriately defined traces.

Interchange Rings

An interchange ring,(R,+,*)is an abelian group with a second binary operation defined so that the interchange law (x+y)*(u+v)=(x*u)+(y*v)holds. An interchange near ring is the same structure based on a group which may not be abelian. It is shown that given any group,G,each interchange(near)ring based on that group is formed from a pair of endomorphisms of G whose images commute, and that all interchange (near)rings based on G can be characterized in this manner. To obtain an associative interchange ring, the endomorphisms must be commuting idempotents in the endomorphism semigroup of G. When G…

Strong inner inverses in endomorphism rings of vector spaces

For $V$ a vector space over a field, or more generally, over a division ring, it is well-known that every $x\in\mathrm{End}(V)$ has an <i>inner inverse</i>, i.e., an element $y\in\mathrm{End}(V)$ satisfying $xyx=x.$ We show here that a large class of such $x$ have inner inverses $y$ that satisfy with $x$ an infinite family of additional monoid relations, making the monoid generated by $x$ and $y$ what is known as an <i>inverse monoid</i> (definition recalled). We obtain consequences of these relations, and related results. P. Nielsen and J. Šter, in a paper to appear, show that a much larger c…

Structure of centralizer algebras

Given an $n\times n$ matrix $c$ over a unitary ring $R$, the centralizer of $c$ in the full $n\times n$ matrix ring $M_n(R)$ is called a principal centralizer matrix ring, denoted by $S_n(c,R)$. We investigate its structure and prove: $(1)$ If $c$ is an invertible matrix with a $c$-free point, or if $R$ has no zero-divisors and $c$ is a Jordan-similar matrix with all eigenvalues in the center of $R$, then $M_n(R)$ is a separable Frobenius extension of $S_{n}(c,R)$ in the sense of Kasch. $(2)$ If $R$ is an integral domain and $c$ is a Jordan-similar matrix, then $S_n(c,R)$ is a cellular $R$-alg…

Centralizers and Inverses to Induction as Equivalence of Categories

Given a ring homomorphism $B \to A$, consider its centralizer $R = A^B$, bimodule endomorphism ring $S = \End {}_BA_B$ and sub-tensor-square ring $T = (A ø_B A)^B$. Nonassociative tensoring by the cyclic modules $R_T$ or ${}_SR$ leads to an equivalence of categories inverse to the functors of induction of restricted $A$-modules or restricted coinduction of $B$-modules in case $A \| B$ is separable, H-separable, split or left depth two (D2). If $R_T$ or ${}_SR$ are projective, this property characterizes separability or splitness for a ring extension. Only in the case of H-separability is $R_T$…

Whitehead Groups of Localizations and the Endomorphism Class Group

We compute the Whitehead groups of the associative rings in a class which includes (twisted) formal power series rings and the augmentation localizations of group rings and polynomial rings. For any associative ring A, we obtain an invariant of a pair (P,α) where P is a finitely generated projective A-module and α:P \to P is an endomorphism. This invariant determines (P,α) up to extensions, yielding a computation of the (reduced) endomorphism class group of A. We also refine the analysis by Pajitnov and Ranicki of the Whitehead group of the Novikov ring.

Centers of categorified endomorphism rings

We prove that for a large class of well-behaved cocomplete categories $\mathcal C$ the weak and strong Drinfeld centers of the monoidal category $\mathcal{E}$ of cocontinuous endofunctors of $\mathcal{C}$ coincide. This generalizes similar results in the literature, where $\mathcal{C}$ is the category of modules over a ring $A$ and hence $\mathcal{E}$ is the category of $A$-bimodules.

常微分算子中心化子的有效计算

Effective computation of centralizers of ODOs

本文研究常微分算子中心化子的计算，提出一种结合 Goodearl 成果与 stationary Gelfand-Dickey 体系求解的算法来计算其基，并开发了生成具有非平凡中心化子的算子族的方法。

对 Leroy 和 Matczuk 关于具有大像的半素左 Noetherian 环自同态单射性问题的肯定回答

Affirmative answer to the Question of Leroy and Matczuk on injectivity of endomorphisms of semiprime left Noetherian rings with large images

本文对 Leroy 和 Matczuk 提出的关于具有大像的半素左 Noetherian 环自同态是否必为单射的问题给出了肯定回答，并证明了更一般的二分法定理及具有 Krull 维数的半素环的相关结论。

Stable finiteness of endomorphism rings

We combine a combinatorial idea of Benjamin Weiss and some localization theory of Grothendieck categories to give a short and completely self-contained proof of the following recent result of Hanfeng Li and Bingbing Liang: Given a left Noetherian ring $R$ and a sofic group $G$, the group-ring $R[G]$ is stably finite.

交换环上的自同态代数与自张量积中的挠元

Endomorphism algebras over commutative rings and torsion in self tensor products

本文研究交换 Noetherian 局部环上有限生成模的自同态环作用下的张量积，证明在温和条件下不可分解模的自张量积必有挠元。

自同态环的中心与自反性

Centers of Endomorphism Rings and Reflexivity

本研究利用模在自同态环及其中心上的自然左模结构，探究有限生成模的同调性质迫使模具有非零自由直和项的条件，给出Lindo经典结论的部分逆命题，并应用于Huneke-Wiegand猜想。

具有类清洁性质的环：自同态、矩阵与结构定理

Rings with Clean-Like Properties: Endomorphism, Matrix and Structural Theorems

研究任意环、阿贝尔群自同态环和有限域上矩阵环的三种类清洁性质，给出弱强k-零清洁、拟2-零清洁和拟3-零清洁的判定条件，改进了多个已有结果。