In [2], Fuchs and Viljoen introduced and classified the $B^{*}$-modules for a valuation ring R: an R-module M is a $B^{*}$-module if $Ex{t}_R^1(M,X)=0$ for each divisible module X and each torsion module X with bounded order. The concept of a $B^{*}$-module was extended to the setting of a torsion theory over an associative ring in [14]. In the present paper, we use categorical methods to investigate the $B^{*}$-modules for a group graded ring. Our most complete result (Theorem 4.10) characterizes $B^{*}$-modules for a strongly graded ring R...

We study the weak dimension of a group-graded ring using methods developed in [B1], [Q] and [R]. We prove that if R is a G-graded ring with G locally finite and the order of every subgroup of G is invertible in R, then the graded weak dimension of R is equal to the ungraded one.

* Partially supported by Universita` di Bari: progetto “Strutture algebriche, geometriche e descrizione degli invarianti ad esse associate”.We compute the cocharacter sequence and generators of the ideal of the weak polynomial identities of the superalgebra M1,1 (E).

Let $R$ be a unital $*$-ring. For any $a,s,t,v,w\in R$ we define the weighted $w$-core inverse and the weighted dual $s$-core inverse, extending the $w$-core inverse and the dual $s$-core inverse, respectively. An element $a\in R$ has a weighted $w$-core inverse with the weight $v$ if there exists some $x\in R$ such that $awxvx=x$, $xvawa=a$ and ${\left(awx\right)}^{*}=awx$. Dually, an element $a\in R$ has a weighted dual $s$-core inverse with the weight $t$ if there exists some $y\in R$ such that $ytysa=y$, $asaty=a$ and ${\left(ysa\right)}^{*}=ysa$. Several characterizations of weighted $w$-core invertible and weighted dual $s$-core invertible...

2000 Mathematics Subject Classification: 13N15, 13A50, 16W25.By using classical invariant theory approach, formulas for computation of the Poincaré series of the kernel of linear locally nilpotent derivations are found.