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1. CMB Online first

Koşan, Tamer; Sahinkaya, Serap; Zhou, Yiqiang
Additive maps on units of rings
Let $R$ be a ring. A map $f: R\rightarrow R$ is additive if $f(a+b)=f(a)+f(b)$ for all elements $a$ and $b$ of $R$. Here a map $f: R\rightarrow R$ is called unit-additive if $f(u+v)=f(u)+f(v)$ for all units $u$ and $v$ of $R$. Motivated by a recent result of Xu, Pei and Yi showing that, for any field $F$, every unit-additive map of ${\mathbb M}_n(F)$ is additive for all $n\ge 2$, this paper is about the question when every unit-additive map of a ring is additive. It is proved that every unit-additive map of a semilocal ring $R$ is additive if and only if either $R$ has no homomorphic image isomorphic to $\mathbb Z_2$ or $R/J(R)\cong \mathbb Z_2$ with $2=0$ in $R$. Consequently, for any semilocal ring $R$, every unit-additive map of ${\mathbb M}_n(R)$ is additive for all $n\ge 2$. These results are further extended to rings $R$ such that $R/J(R)$ is a direct product of exchange rings with primitive factors Artinian. A unit-additive map $f$ of a ring $R$ is called unit-homomorphic if $f(uv)=f(u)f(v)$ for all units $u,v$ of $R$. As an application, the question of when every unit-homomorphic map of a ring is an endomorphism is addressed.

Keywords:additive map, unit, 2-sum property, semilocal ring, exchange ring with primitive factors Artinian
Categories:15A99, 16U60, 16L30

2. CMB Online first

Bavula, V. V.; Lu, T.
Classification of simple weight modules over the Schrödinger algebra
A classification of simple weight modules over the Schrödinger algebra is given. The Krull and the global dimensions are found for the centralizer $C_{\mathcal{S}}(H)$ (and some of its prime factor algebras) of the Cartan element $H$ in the universal enveloping algebra $\mathcal{S}$ of the Schrödinger (Lie) algebra. The simple $C_{\mathcal{S}}(H)$-modules are classified. The Krull and the global dimensions are found for some (prime) factor algebras of the algebra $\mathcal{S}$ (over the centre). It is proved that some (prime) factor algebras of $\mathcal{S}$ and $C_{\mathcal{S}}(H)$ are tensor homological/Krull minimal.

Keywords:weight module, simple module, centralizer, Krull dimension, global dimension, tensor homological minimal algebra, tensor Krull minimal algebra
Categories:17B10, 17B20, 17B35, 16E10, 16P90, 16P40, 16P50

3. CMB Online first

Buijs, Urtzi; Félix, Yves; Murillo, Aniceto; Tanré, Daniel
Maurer-Cartan elements in the Lie models of finite simplicial complexes
In a previous work, we have associated a complete differential graded Lie algebra to any finite simplicial complex in a functorial way. Similarly, we have also a realization functor from the category of complete differential graded Lie algebras to the category of simplicial sets. We have already interpreted the homology of a Lie algebra in terms of homotopy groups of its realization. In this paper, we begin a dictionary between models and simplicial complexes by establishing a correspondence between the Deligne groupoid of the model and the connected components of the finite simplicial complex.

Keywords:complete differential graded Lie algebra, Maurer-Cartan element, rational homotopy theory
Category:16E45

4. CMB Online first

Lee, Tsiu-Kwen
Ad-nilpotent elements of semiprime rings with involution
Let $R$ be an $n!$-torsion free semiprime ring with involution $*$ and with extended centroid $C$, where $n\gt 1$ is a positive integer. We characterize $a\in K$, the Lie algebra of skew elements in $R$, satisfying $(\operatorname{ad}_a)^n=0$ on $K$. This generalizes both Martindale and Miers' theorem and the theorem of Brox et al. To prove it we first prove that if $a, b\in R$ satisfy $(\operatorname{ad}_a)^n=\operatorname{ad}_b$ on $R$, where either $n$ is even or $b=0$, then $\big(a-\lambda\big)^{[\frac{n+1}{2}]}=0$ for some $\lambda\in C$.

Keywords:Semiprime ring, Lie algebra, Jordan algebra, faithful $f$-free, involution, skew (symmetric) element, ad-nilpotent element, Jordan element
Categories:16N60, 16W10, 17B60

5. CMB Online first

Khojasteh, Sohiela; Nikmehr, Mohammad Javad
The Weakly Nilpotent Graph of a Commutative Ring
Let $R$ be a commutative ring with non-zero identity. In this paper, we introduced the weakly nilpotent graph of a commutative ring. The weakly nilpotent graph of $R$ is denoted by $\Gamma_w(R)$ is a graph with the vertex set $R^{*}$ and two vertices $x$ and $y$ are adjacent if and only if $xy\in N(R)^{*}$, where $R^{*}=R\setminus\{0\}$ and $N(R)^{*}$ is the set of all non-zero nilpotent elements of $R$. In this article, we determine the diameter of weakly nilpotent graph of an Artinian ring. We prove that if $\Gamma_w(R)$ is a forest, then $\Gamma_w(R)$ is a union of a star and some isolated vertices. We study the clique number, the chromatic number and the independence number of $\Gamma_w(R)$. Among other results, we show that for an Artinian ring $R$, $\Gamma_w(R)$ is not a disjoint union of cycles or a unicyclic graph. For Artinan ring, we determine $\operatorname{diam}(\overline{\Gamma_w(R)})$. Finally, we characterize all commutative rings $R$ for which $\overline{\Gamma_w(R)}$ is a cycle, where $\overline{\Gamma_w(R)}$ is the complement of the weakly nilpotent graph of $R$.

Keywords:weakly nilpotent graph, zero-divisor graph, diameter, girth
Categories:05C15, 16N40, 16P20

6. CMB Online first

Wang, Long; Castro-Gonzalez, Nieves; Chen, Jianlong
Characterizations of outer generalized inverses
Let $R$ be a ring and $b, c\in R$. In this paper, we give some characterizations of the $(b,c)$-inverse, in terms of the direct sum decomposition, the annihilator and the invertible elements. Moreover, elements with equal $(b,c)$-idempotents related to their $(b, c)$-inverses are characterized, and the reverse order rule for the $(b,c)$-inverse is considered.

Keywords:$(b, c)$-inverse, $(b, c)$-idempotent, regularity, image-kernel $(p, q)$-inverse, ring
Categories:15A09, 16U99

7. CMB Online first

Eroǧlu, Münevver Pınar; Argaç, Nurcan
On Identities with Composition of Generalized Derivations
Let $R$ be a prime ring with extended centroid $C$, $Q$ maximal right ring of quotients of $R$, $RC$ central closure of $R$ such that $dim_{C}(RC) \gt 4$, $f(X_{1},\dots,X_{n})$ a multilinear polynomial over $C$ which is not central-valued on $R$ and $f(R)$ the set of all evaluations of the multilinear polynomial $f\big(X_{1},\dots,X_{n}\big)$ in $R$. Suppose that $G$ is a nonzero generalized derivation of $R$ such that $G^2\big(u\big)u \in C$ for all $u\in f(R)$ then one of the following conditions holds: (I) there exists $a\in Q$ such that $a^2=0$ and either $G(x)=ax$ for all $x\in R$ or $G(x)=xa$ for all $x\in R$; (II) there exists $a\in Q$ such that $0\neq a^2\in C$ and either $G(x)=ax$ for all $x\in R$ or $G(x)=xa$ for all $x\in R$ and $f(X_{1},\dots,X_{n})^{2}$ is central-valued on $R$; (III) $char(R)=2$ and one of the following holds: (i) there exist $a, b\in Q$ such that $G(x)=ax+xb$ for all $x\in R$ and $a^{2}=b^{2}\in C$; (ii) there exist $a, b\in Q$ such that $G(x)=ax+xb$ for all $x\in R$, $a^{2}, b^{2}\in C$ and $f(X_{1},\ldots,X_{n})^{2}$ is central-valued on $R$; (iii) there exist $a \in Q$ and an $X$-outer derivation $d$ of $R$ such that $G(x)=ax+d(x)$ for all $x\in R$, $d^2=0$ and $a^2+d(a)=0$; (iv) there exist $a \in Q$ and an $X$-outer derivation $d$ of $R$ such that $G(x)=ax+d(x)$ for all $x\in R$, $d^2=0$, $a^2+d(a)\in C$ and $f(X_{1},\dots,X_{n})^{2}$ is central-valued on $R$. Moreover, we characterize the form of nonzero generalized derivations $G$ of $R$ satisfying $G^2(x)=\lambda x$ for all $x\in R$, where $\lambda \in C$.

Keywords:prime ring, generalized derivation, composition, extended centroid, multilinear polynomial, maximal right ring of quotients
Categories:16N60, 16N25

8. CMB 2016 (vol 60 pp. 3)

Akbari, Saeeid; Alilou, Abbas; Amjadi, Jafar; Sheikholeslami, Seyed Mahmoud
The Co-annihilating-ideal Graphs of Commutative Rings
Let $R$ be a commutative ring with identity. The co-annihilating-ideal graph of $R$, denoted by $\mathcal{A}_R$, is a graph whose vertex set is the set of all non-zero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent whenever ${\operatorname {Ann}}(I)\cap {\operatorname {Ann}}(J)=\{0\}$. In this paper we initiate the study of the co-annihilating ideal graph of a commutative ring and we investigate its properties.

Keywords:commutative ring, co-annihilating ideal graph
Categories:13A15, 16N40

9. CMB 2016 (vol 59 pp. 794)

Hashemi, Ebrahim; Amirjan, R.
Zero-divisor Graphs of Ore Extensions over Reversible Rings
Let $R$ be an associative ring with identity. First we prove some results about zero-divisor graphs of reversible rings. Then we study the zero-divisors of the skew power series ring $R[[x;\alpha]]$, whenever $R$ is reversible and $\alpha$-compatible. Moreover, we compare the diameter and girth of the zero-divisor graphs of $\Gamma(R)$, $\Gamma(R[x;\alpha,\delta])$ and $\Gamma(R[[x;\alpha]])$, when $R$ is reversible and $(\alpha,\delta)$-compatible.

Keywords:zero-divisor graphs, reversible rings, McCoy rings, polynomial rings, power series rings
Categories:13B25, 05C12, 16S36

10. CMB 2016 (vol 59 pp. 652)

Su, Huadong
On the Diameter of Unitary Cayley Graphs of Rings
The unitary Cayley graph of a ring $R$, denoted $\Gamma(R)$, is the simple graph defined on all elements of $R$, and where two vertices $x$ and $y$ are adjacent if and only if $x-y$ is a unit in $R$. The largest distance between all pairs of vertices of a graph $G$ is called the diameter of $G$, and is denoted by ${\rm diam}(G)$. It is proved that for each integer $n\geq1$, there exists a ring $R$ such that ${\rm diam}(\Gamma(R))=n$. We also show that ${\rm diam}(\Gamma(R))\in \{1,2,3,\infty\}$ for a ring $R$ with $R/J(R)$ self-injective and classify all those rings with ${\rm diam}(\Gamma(R))=1$, 2, 3 and $\infty$, respectively.

Keywords:unitary Cayley graph, diameter, $k$-good, unit sum number, self-injective ring
Categories:05C25, 16U60, 05C12

11. CMB 2016 (vol 59 pp. 461)

Ara, Pere; O'Meara, Kevin C.
The Nilpotent Regular Element Problem
We use George Bergman's recent normal form for universally adjoining an inner inverse to show that, for general rings, a nilpotent regular element $x$ need not be unit-regular. This contrasts sharply with the situation for nilpotent regular elements in exchange rings (a large class of rings), and for general rings when all powers of the nilpotent element $x$ are regular.

Keywords:nilpotent element, von Neumann regular element, unit-regular, Bergman's normal form
Categories:16E50, 16U99, 16S10, 16S15

12. CMB 2016 (vol 59 pp. 661)

Ying, Zhiling; Koşan, Tamer; Zhou, Yiqiang
Rings in Which Every Element is a Sum of Two Tripotents
Let $R$ be a ring. The following results are proved: $(1)$ every element of $R$ is a sum of an idempotent and a tripotent that commute iff $R$ has the identity $x^6=x^4$ iff $R\cong R_1\times R_2$, where $R_1/J(R_1)$ is Boolean with $U(R_1)$ a group of exponent $2$ and $R_2$ is zero or a subdirect product of $\mathbb Z_3$'s; $(2)$ every element of $R$ is either a sum or a difference of two commuting idempotents iff $R\cong R_1\times R_2$, where $R_1/J(R_1)$ is Boolean with $J(R_1)=0$ or $J(R_1)=\{0,2\}$, and $R_2$ is zero or a subdirect product of $\mathbb Z_3$'s; $(3)$ every element of $R$ is a sum of two commuting tripotents iff $R\cong R_1\times R_2\times R_3$, where $R_1/J(R_1)$ is Boolean with $U(R_1)$ a group of exponent $2$, $R_2$ is zero or a subdirect product of $\mathbb Z_3$'s, and $R_3$ is zero or a subdirect product of $\mathbb Z_5$'s.

Keywords:idempotent, tripotent, Boolean ring, polynomial identity $x^3=x$, polynomial identity $x^6=x^4$, polynomial identity $x^8=x^4$
Categories:16S50, 16U60, 16U90

13. CMB 2016 (vol 59 pp. 340)

Kȩpczyk, Marek
A Note on Algebras that are Sums of Two Subalgebras
We study an associative algebra $A$ over an arbitrary field, that is a sum of two subalgebras $B$ and $C$ (i.e. $A=B+C$). We show that if $B$ is a right or left Artinian $PI$ algebra and $C$ is a $PI$ algebra, then $A$ is a $PI$ algebra. Additionally we generalize this result for semiprime algebras $A$. Consider the class of all semisimple finite dimensional algebras $A=B+C$ for some subalgebras $B$ and $C$ which satisfy given polynomial identities $f=0$ and $g=0$, respectively. We prove that all algebras in this class satisfy a common polynomial identity.

Keywords:rings with polynomial identities, prime rings
Categories:16N40, 16R10, , 16S36, 16W60, 16R20

14. CMB 2016 (vol 59 pp. 271)

Dehghani-Zadeh, Fatemeh
Artinianness of Composed Graded Local Cohomology Modules
Let $R=\bigoplus_{n\geq0}R_{n}$ be a graded Noetherian ring with local base ring $(R_{0}, \mathfrak{m}_{0})$ and let $R_{+}=\bigoplus_{n\gt 0}R_{n}$, $M$ and $N$ be finitely generated graded $R$-modules and $\mathfrak{a}=\mathfrak{a}_{0}+R_{+}$ an ideal of $R$. We show that $H^{j}_{\mathfrak{b}_{0}}(H^{i}_{\mathfrak{a}}(M,N))$ and $H^{i}_{\mathfrak{a}}(M, N)/\mathfrak{b}_{0}H^{i}_{\mathfrak{a}}(M,N)$ are Artinian for some $i^{,}s$ and $j^{,}s$ with a specified property, where $\mathfrak{b}_{o}$ is an ideal of $R_{0}$ such that $\mathfrak{a}_{0}+\mathfrak{b}_{0}$ is an $\mathfrak{m}_{0}$-primary ideal.

Keywords:generalized local cohomology, Artinian, graded module
Categories:13D45, 13E10, 16W50

15. CMB 2016 (vol 59 pp. 258)

De Filippis, Vincenzo
Annihilators and Power Values of Generalized Skew Derivations on Lie Ideals
Let $R$ be a prime ring of characteristic different from $2$, $Q_r$ be its right Martindale quotient ring and $C$ be its extended centroid. Suppose that $F$ is a generalized skew derivation of $R$, $L$ a non-central Lie ideal of $R$, $0 \neq a\in R$, $m\geq 0$ and $n,s\geq 1$ fixed integers. If \[ a\biggl(u^mF(u)u^n\biggr)^s=0 \] for all $u\in L$, then either $R\subseteq M_2(C)$, the ring of $2\times 2$ matrices over $C$, or $m=0$ and there exists $b\in Q_r$ such that $F(x)=bx$, for any $x\in R$, with $ab=0$.

Keywords:generalized skew derivation, prime ring
Categories:16W25, 16N60

16. CMB 2015 (vol 58 pp. 741)

Gao, Zenghui
Homological Properties Relative to Injectively Resolving Subcategories
Let $\mathcal{E}$ be an injectively resolving subcategory of left $R$-modules. A left $R$-module $M$ (resp. right $R$-module $N$) is called $\mathcal{E}$-injective (resp. $\mathcal{E}$-flat) if $\operatorname{Ext}_R^1(G,M)=0$ (resp. $\operatorname{Tor}_1^R(N,G)=0$) for any $G\in\mathcal{E}$. Let $\mathcal{E}$ be a covering subcategory. We prove that a left $R$-module $M$ is $\mathcal{E}$-injective if and only if $M$ is a direct sum of an injective left $R$-module and a reduced $\mathcal{E}$-injective left $R$-module. Suppose $\mathcal{F}$ is a preenveloping subcategory of right $R$-modules such that $\mathcal{E}^+\subseteq\mathcal{F}$ and $\mathcal{F}^+\subseteq\mathcal{E}$. It is shown that a finitely presented right $R$-module $M$ is $\mathcal{E}$-flat if and only if $M$ is a cokernel of an $\mathcal{F}$-preenvelope of a right $R$-module. In addition, we introduce and investigate the $\mathcal{E}$-injective and $\mathcal{E}$-flat dimensions of modules and rings. We also introduce $\mathcal{E}$-(semi)hereditary rings and $\mathcal{E}$-von Neumann regular rings and characterize them in terms of $\mathcal{E}$-injective and $\mathcal{E}$-flat modules.

Keywords:injectively resolving subcategory, \mathcal{E}-injective module (dimension), \mathcal{E}-flat module (dimension), cover, preenvelope, \mathcal{E}-(semi)hereditary ring
Categories:16E30, 16E10, 16E60

17. CMB 2015 (vol 58 pp. 730)

Efrat, Ido; Matzri, Eliyahu
Vanishing of Massey Products and Brauer Groups
Let $p$ be a prime number and $F$ a field containing a root of unity of order $p$. We relate recent results on vanishing of triple Massey products in the mod-$p$ Galois cohomology of $F$, due to Hopkins, Wickelgren, Mináċ, and Tân, to classical results in the theory of central simple algebras. For global fields, we prove a stronger form of the vanishing property.

Keywords:Galois cohomology, Brauer groups, triple Massey products, global fields
Categories:16K50, 11R34, 12G05, 12E30

18. CMB 2015 (vol 58 pp. 233)

Bergen, Jeffrey
Affine Actions of $U_q(sl(2))$ on Polynomial Rings
We classify the affine actions of $U_q(sl(2))$ on commutative polynomial rings in $m \ge 1$ variables. We show that, up to scalar multiplication, there are two possible actions. In addition, for each action, the subring of invariants is a polynomial ring in either $m$ or $m-1$ variables, depending upon whether $q$ is or is not a root of $1$.

Keywords:skew derivation, quantum group, invariants
Categories:16T20, 17B37, 20G42

19. CMB 2015 (vol 58 pp. 263)

De Filippis, Vincenzo; Mamouni, Abdellah; Oukhtite, Lahcen
Generalized Jordan Semiderivations in Prime Rings
Let $R$ be a ring, $g$ an endomorphism of $R$. The additive mapping $d\colon R\rightarrow R$ is called Jordan semiderivation of $R$, associated with $g$, if $$d(x^2)=d(x)x+g(x)d(x)=d(x)g(x)+xd(x)\quad \text{and}\quad d(g(x))=g(d(x))$$ for all $x\in R$. The additive mapping $F\colon R\rightarrow R$ is called generalized Jordan semiderivation of $R$, related to the Jordan semiderivation $d$ and endomorphism $g$, if $$F(x^2)=F(x)x+g(x)d(x)=F(x)g(x)+xd(x)\quad \text{and}\quad F(g(x))=g(F(x))$$ for all $x\in R$. In the present paper we prove that if $R$ is a prime ring of characteristic different from $2$, $g$ an endomorphism of $R$, $d$ a Jordan semiderivation associated with $g$, $F$ a generalized Jordan semiderivation associated with $d$ and $g$, then $F$ is a generalized semiderivation of $R$ and $d$ is a semiderivation of $R.$ Moreover, if $R$ is commutative then $F=d$.

Keywords:semiderivation, generalized semiderivation, Jordan semiderivation, prime ring
Category:16W25

20. CMB 2014 (vol 58 pp. 134)

Nasseh, Saeed
On the Generalized Auslander-Reiten Conjecture under Certain Ring Extensions
We show under some conditions that a Gorenstein ring $R$ satisfies the Generalized Auslander-Reiten Conjecture if and only if so does $R[x]$. When $R$ is a local ring we prove the same result for some localizations of $R[x]$.

Keywords:Auslander-Reiten conjecture, finitistic extension degree, Gorenstein rings
Categories:13D07, 16E30, 16E65

21. CMB Online first

Nasseh, Saeed
On the Generalized Auslander-Reiten Conjecture under Certain Ring Extensions
We show under some conditions that a Gorenstein ring $R$ satisfies the Generalized Auslander-Reiten Conjecture if and only if so does $R[x]$. When $R$ is a local ring we prove the same result for some localizations of $R[x]$.

Keywords:Auslander-Reiten conjecture, finitistic extension degree, Gorenstein rings
Categories:13D07, 16E30, 16E65

22. CMB 2014 (vol 57 pp. 814)

Hou, Ruchen
On Global Dimensions of Tree Type Finite Dimensional Algebras
A formula is provided to explicitly describe global dimensions of all kinds of tree type finite dimensional $k-$algebras for $k$ an algebraic closed field. In particular, it is pointed out that if the underlying tree type quiver has $n$ vertices, then the maximum of possible global dimensions is $n-1$.

Keywords:global dimension, tree type finite dimensional $k-$algebra, quiver
Categories:16D40, 16E10, , 16G20

23. CMB 2014 (vol 57 pp. 609)

Nasr-Isfahani, Alireza
Jacobson Radicals of Skew Polynomial Rings of Derivation Type
We provide necessary and sufficient conditions for a skew polynomial ring of derivation type to be semiprimitive, when the base ring has no nonzero nil ideals. This extends existing results on the Jacobson radical of skew polynomial rings of derivation type.

Keywords:skew polynomial rings, Jacobson radical, derivation
Categories:16S36, 16N20

24. CMB 2014 (vol 57 pp. 511)

Gonçalves, Daniel
Simplicity of Partial Skew Group Rings of Abelian Groups
Let $A$ be a ring with local units, $E$ a set of local units for $A$, $G$ an abelian group and $\alpha$ a partial action of $G$ by ideals of $A$ that contain local units. We show that $A\star_{\alpha} G$ is simple if and only if $A$ is $G$-simple and the center of the corner $e\delta_0 (A\star_{\alpha} G) e \delta_0$ is a field for all $e\in E$. We apply the result to characterize simplicity of partial skew group rings in two cases, namely for partial skew group rings arising from partial actions by clopen subsets of a compact set and partial actions on the set level.

Keywords:partial skew group rings, simple rings, partial actions, abelian groups
Categories:16S35, 37B05

25. CMB 2014 (vol 57 pp. 264)

Dai, Li; Dong, Jingcheng
On Semisimple Hopf Algebras of Dimension $pq^n$
Let $p,q$ be prime numbers with $p^2\lt q$, $n\in \mathbb{N}$, and $H$ a semisimple Hopf algebra of dimension $pq^n$ over an algebraically closed field of characteristic $0$. This paper proves that $H$ must possess one of the following structures: (1) $H$ is semisolvable; (2) $H$ is a Radford biproduct $R\# kG$, where $kG$ is the group algebra of group $G$ of order $p$, and $R$ is a semisimple Yetter--Drinfeld Hopf algebra in ${}^{kG}_{kG}\mathcal{YD}$ of dimension $q^n$.

Keywords:semisimple Hopf algebra, semisolvability, Radford biproduct, Drinfeld double
Category:16W30
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