In this paper, we study the images of multilinear graded polynomials on the graded algebra of upper triangular matrices $UT_n$. For positive integers $q\leq n$, we classify these images on $UT_{n}$ endowed with a particular elementary ${\mathbb {Z}}_{q}$-grading. As a consequence, we obtain the images of multilinear graded polynomials on $UT_{n}$ with the natural ${\mathbb {Z}}_{n}$-grading. We apply this classification in order to give a new condition for a multilinear polynomial in terms of graded identities so that to obtain the traceless matrices in its image on the full matrix algebra. We also describe the images of multilinear polynomials on the graded algebras $UT_{2}$ and $UT_{3}$, for arbitrary gradings. We finish the paper by proving a similar result for the graded Jordan algebra $UJ_{2}$, and also for $UJ_{3}$ endowed with the natural elementary ${\mathbb {Z}}_{3}$-grading.

Let F be a field of characteristic zero, and let $UT_2$ be the algebra of $2 \times 2$ upper triangular matrices over F. In a previous paper by Centrone and Yasumura, the authors give a description of the action of Taft’s algebras $H_m$ on $UT_2$ and its $H_m$ -identities. In this paper, we give a complete description of the space of multilinear $H_m$ -identities in the language of Young diagrams through the representation theory of the hyperoctahedral group. We finally prove that the variety of $H_m$ -module algebras generated by $UT_2$ has the Specht property, i.e., every $T^{H_m}$ -ideal containing the $H_m$ -identities of $UT_2$ is finitely based.

Let $F$ be an infinite field of positive characteristic $p > 2$ and let $G$ be a group. In this paper, we study the graded identities satisfied by an associative algebra equipped with an elementary $G$-grading. Let $E$ be the infinite-dimensional Grassmann algebra. For every $a$, $b\in \mathbb {N}$, we provide a basis for the graded polynomial identities, up to graded monomial identities, for the verbally prime algebras $M_{a,b}(E)$, as well as their tensor products, with their elementary gradings. Moreover, we give an alternative proof of the fact that the tensor product $M_{a,b}(E)\otimes M_{r,s}(E)$ and $M_{ar+bs,as+br}(E)$ are $F$-algebras which are not PI equivalent. Actually, we prove that the $T_{G}$-ideal of the former algebra is contained in the $T$-ideal of the latter. Furthermore, the inclusion is proper. Recall that it is well known that these algebras satisfy the same multilinear identities and hence in characteristic 0 they are PI equivalent.

The main result of this note implies that any function from the product of several vector spaces to a vector space can be uniquely decomposed into the sum of mutually orthogonal functions that are odd in some of the arguments and even in the other arguments. Probabilistic notions and facts are employed to simplify statements and proofs.

Let $R$ be a semiprime ring with center $Z\left( R \right)$. For $x,\,y\,\in \,R$, we denote by $\left[ x,\,y \right]\,=\,xy\,-\,yx$ the commutator of $x$ and $y$. If $\sigma $ is a non-identity automorphism of $R$ such that

1

$$\left[ \left[ \cdot \cdot \cdot \,\left[ \left[ \sigma \left( {{x}^{n0}} \right),\,{{x}^{n1}} \right],\,{{x}^{n2}} \right],\cdot \cdot \cdot \right],\,{{x}^{nk}} \right]\,=\,0$$

for all $x\,\in \,R$, where ${{n}_{0}},\,{{n}_{1}},\,{{n}_{2}},\,...,\,{{n}_{k}}$ are fixed positive integers, then there exists a map $\mu \,:\,R\,\to \,Z\left( R \right)$ such that $\sigma \left( x \right)\,=\,x\,+\,\mu \left( x \right)$ for all $x\,\in \,R$. In particular, when $R$ is a prime ring, $R$ is commutative.

Let F be a field of characteristic p ≠ 2 and G a group without 2-elements having an involution ∗. Extend the involution linearly to the group ring FG, and let (FG)− denote the set of skew elements with respect to ∗. In this paper, we show that if G is finite and (FG)− is Lie metabelian, then G is nilpotent. Based on this result, we deduce that if G is torsion, p > 7 and (FG)− is Lie metabelian, then G must be abelian. Exceptions are constructed for smaller values of p.

Let $F$ be an algebraically closed field of characteristic zero, and let $A$ be an associative unitary $F$-algebra graded by a group of prime order. We prove that if $A$ is finite dimensional then the graded exponent of $A$ exists and is an integer.

Let $R$ be an associative ring with unity. Then $R$ is said to be a right McCoy ring when the equation $f\left( x \right)g\left( x \right)\,=\,0$ (over $R\left[ x \right]$), where $0\,\ne \,f\left( x \right)$, $g\left( x \right)\,\in \,R\left[ x \right]$, implies that there exists a nonzero element $c\,\in \,R$ such that $f\left( x \right)c\,=\,0$. In this paper, we characterize some basic ring extensions of right McCoy rings and we prove that if $R$ is a right McCoy ring, then $R\left[ x \right]/\left( {{x}^{n}} \right)$ is a right McCoy ring for any positive integer $n\,\ge \,2$.

We study group algebras $FG$ which can be graded by a finite abelian group $\Gamma $ such that $FG$ is $\beta $-commutative for a skew-symmetric bicharacter $\beta $ on $\Gamma $ with values in ${{F}^{*}}$.

Let $\mathbb{K}$ be a field of characteristic zero, and $*\,=\,t$ the transpose involution for the matrix algebra ${{M}_{2}}\left( \mathbb{K} \right)$. Let $\mathfrak{U}$ be a proper subvariety of the variety of algebras with involution generated by $\left( {{M}_{2}}\left( \mathbb{K} \right),\,* \right)$. We define two sequences of algebras with involution ${{R}_{p}},\,{{S}_{q}}$, where $p,\,q\,\in \,\mathbb{N}$. Then we show that ${{T}_{*}}\left( \mathfrak{U} \right)$ and ${{T}_{*}}\left( {{R}_{p}}\oplus \,{{S}_{q}} \right)$ are $*$-asymptotically equivalent for suitable $p,\,q$.

Let ${{M}_{n}}(F)$ be the algebra of $n\times n$ matrices over a field $F$ of characteristic $p>2$ and let $*$ be an involution on ${{M}_{n}}(F)$. If ${{s}_{1}},...,{{s}_{r}}$ are symmetric variables we determine the smallest $r$ such that the polynomial

$${{P}_{r}}({{S}_{1}},...,{{S}_{r}})\,=\,\sum\limits_{\sigma \in {{S}_{r}}}{{{S}_{\sigma (1)}}...{{S}_{\sigma (r)}}}$$

is a $*$-polynomial identity of ${{M}_{n}}(F)$ under either the symplectic or the transpose involution. We also prove an analogous result for the polynomial

$${{C}_{r}}\left( {{k}_{1}},...,{{k}_{r,}}{{{{k}'}}_{1}},...,{{{{k}'}}_{r}} \right)=\sum\limits_{\sigma ,\tau \in {{S}_{r}}}{{{k}_{\sigma \left( 1 \right)}}{{{{k}'}}_{\tau \left( 1 \right)}}\cdot \cdot \cdot {{k}_{\sigma \left( r \right)}}{{{{k}'}}_{\tau \left( r \right)}}}$$

where ${{k}_{1}},...,{{k}_{r}},k_{1}^{'},...,k_{r}^{'}$ are skew variables under the transpose involution.

Let R be a prime ring with involution and d, δ be derivations on R. Suppose that xd(x)—δ(x)x is central for all symmetric x or for all skew x. Then d = δ = 0 unless R is a commutative integral domain or an order of a 4-dimensional central simple algebra.

Let M2(K, *) be the algebra of 2 × 2 matrices with involution over a field K of characteristic 0. We obtain the exact values of the cocharacters, codimensions and Hilbert series of the *-T-ideal of the polynomial identities for M2(K, *).