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1. CMB Online first

 Monoidal Categories, 2-Traces, and Cyclic Cohomology In this paper we show that to a unital associative algebra object (resp. co-unital co-associative co-algebra object) of any abelian monoidal category $(\mathcal{C}, \otimes)$ endowed with a symmetric $2$-trace, i.e. an $F\in Fun(\mathcal{C}, \operatorname{Vec})$ satisfying some natural trace-like conditions, one can attach a cyclic (resp.cocyclic) module, and therefore speak of the (co)cyclic homology of the (co)algebra with coefficients in $F$". Furthermore, we observe that if $\mathcal{M}$ is a $\mathcal{C}$-bimodule category and $(F, M)$ is a stable central pair, i.e., $F\in Fun(\mathcal{M}, \operatorname{Vec})$ and $M\in \mathcal{M}$ satisfy certain conditions, then $\mathcal{C}$ acquires a symmetric 2-trace. The dual notions of symmetric $2$-contratraces and stable central contrapairs are derived as well. As an application we can recover all Hopf cyclic type (co)homology theories. Keywords:monoidal category, abelian and additive category, cyclic homology, Hopf algebraCategories:16T05, 18D10, 19D55

2. CMB 2018 (vol 61 pp. 688)

Bavula, V. V.; Lu, T.
 The Universal Enveloping Algebra of the SchrÃ¶dinger Algebra and its Prime Spectrum The prime, completely prime, maximal and primitive spectra are classified for the universal enveloping algebra of the SchrÃ¶dinger algebra. For all of these ideals their explicit generators are given. A counterexample is constructed to the conjecture of Cheng and Zhang about non-existence of simple singular Whittaker modules for the SchrÃ¶dinger algebra (and all such modules are classified). It is proved that the conjecture holds 'generically'. Keywords:prime ideal, weight module, simple module, centralizer, Whittaker moduleCategories:17B10, 16D25, 16D60, 16D70, 16P50

3. CMB 2017 (vol 61 pp. 865)

Shaul, Liran
 Homological Dimensions of Local (Co)homology over Commutative DG-rings Let $A$ be a commutative noetherian ring, let $\mathfrak{a}\subseteq A$ be an ideal, and let $I$ be an injective $A$-module. A basic result in the structure theory of injective modules states that the $A$-module $\Gamma_{\mathfrak{a}}(I)$ consisting of $\mathfrak{a}$-torsion elements is also an injective $A$-module. Recently, de Jong proved a dual result: If $F$ is a flat $A$-module, then the $\mathfrak{a}$-adic completion of $F$ is also a flat $A$-module. In this paper we generalize these facts to commutative noetherian DG-rings: let $A$ be a commutative non-positive DG-ring such that $\mathrm{H}^0(A)$ is a noetherian ring, and for each $i\lt 0$, the $\mathrm{H}^0(A)$-module $\mathrm{H}^i(A)$ is finitely generated. Given an ideal $\bar{\mathfrak{a}} \subseteq \mathrm{H}^0(A)$, we show that the local cohomology functor $\mathrm{R}\Gamma_{\bar{\mathfrak{a}}}$ associated to $\bar{\mathfrak{a}}$ does not increase injective dimension. Dually, the derived $\bar{\mathfrak{a}}$-adic completion functor $\mathrm{L}\Lambda_{\bar{\mathfrak{a}}}$ does not increase flat dimension. Keywords:local cohomology, derived completion, homological dimension, commutative DG-ringCategories:13B35, 13D05, 13D45, 16E45

4. CMB 2017 (vol 60 pp. 879)

Zheng, Yuefei; Huang, Zhaoyong
 Triangulated Equivalences Involving Gorenstein Projective Modules For any ring $R$, we show that, in the bounded derived category $D^{b}(\operatorname{Mod} R)$ of left $R$-modules, the subcategory of complexes with finite Gorenstein projective (resp. injective) dimension modulo the subcategory of complexes with finite projective (resp. injective) dimension is equivalent to the stable category $\underline{\mathbf{GP}}(\operatorname{Mod} R)$ (resp. $\overline{\mathbf{GI}}(\operatorname{Mod} R)$) of Gorenstein projective (resp. injective) modules. As a consequence, we get that if $R$ is a left and right noetherian ring admitting a dualizing complex, then $\underline{\mathbf{GP}}(\operatorname{Mod} R)$ and $\overline{\mathbf{GI}}(\operatorname{Mod} R)$ are equivalent. Keywords:triangulated equivalence, Gorenstein projective module, stable category, derived category, homotopy categoryCategories:18G25, 16E35

5. CMB 2017 (vol 61 pp. 225)

Bichon, Julien; Kyed, David; Raum, Sven
 Higher $\ell^2$-Betti Numbers of Universal Quantum Groups We calculate all $\ell^2$-Betti numbers of the universal discrete Kac quantum groups $\hat{\mathrm U}^+_n$ as well as their half-liberated counterparts $\hat{\mathrm U}^*_n$. Keywords:$\ell^2$-Betti number, free unitary quantum group, half-liberated unitary quantum group, free product formula, extensionCategories:16T05, 46L65, 20G42

6. CMB 2017 (vol 61 pp. 301)

Józiak, Paweł
 Remarks on Hopf Images and Quantum Permutation Groups $S_n^+$ Motivated by a question of A. Skalski and P.M. SoÅtan (2016) about inner faithfulness of the S. Curran's map of extending a quantum increasing sequence to a quantum permutation, we revisit the results and techniques of T. Banica and J. Bichon (2009) and study some group-theoretic properties of the quantum permutation group on $4$ points. This enables us not only to answer the aforementioned question in positive in case $n=4, k=2$, but also to classify the automorphisms of $S_4^+$, describe all the embeddings $O_{-1}(2)\subset S_4^+$ and show that all the copies of $O_{-1}(2)$ inside $S_4^+$ are conjugate. We then use these results to show that the converse to the criterion we applied to answer the aforementioned question is not valid. Keywords:Hopf image, quantum permutation group, compact quantum groupCategories:20G42, 81R50, 46L89, 16W35

7. CMB 2017 (vol 61 pp. 16)

Bavula, V. V.; Lu, T.
 Classification of simple weight modules over the SchrÃ¶dinger algebra A classification of simple weight modules over the SchrÃ¶dinger 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 SchrÃ¶dinger (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 algebraCategories:17B10, 17B20, 17B35, 16E10, 16P90, 16P40, 16P50

8. CMB 2017 (vol 61 pp. 130)

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 ArtinianCategories:15A99, 16U60, 16L30

9. CMB 2017 (vol 60 pp. 470)

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 theoryCategory:16E45

10. CMB 2017 (vol 61 pp. 318)

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 elementCategories:16N60, 16W10, 17B60

11. CMB 2017 (vol 60 pp. 319)

 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, girthCategories:05C15, 16N40, 16P20

12. CMB 2017 (vol 60 pp. 861)

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, ringCategories:15A09, 16U99

13. 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 graphCategories:13A15, 16N40

14. CMB 2016 (vol 60 pp. 721)

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 quotientsCategories:16N60, 16N25

15. 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 ringsCategories:13B25, 05C12, 16S36

16. CMB 2016 (vol 59 pp. 652)

 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 ringCategories:05C25, 16U60, 05C12

17. 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 formCategories:16E50, 16U99, 16S10, 16S15

18. 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

19. CMB 2016 (vol 59 pp. 271)

 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 moduleCategories:13D45, 13E10, 16W50

20. 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 ringsCategories:16N40, 16R10, , 16S36, 16W60, 16R20

21. 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 ringCategories:16W25, 16N60

22. 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 ringCategories:16E30, 16E10, 16E60

23. 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, MinÃ¡Ä, and TÃ¢n, 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 fieldsCategories:16K50, 11R34, 12G05, 12E30

24. 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, invariantsCategories:16T20, 17B37, 20G42

25. 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 ringCategory:16W25
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