We define an operator α on ℂ³ ⊗ ℂ³ associated with the quantum group $U_q\left(2\right)$, which satisfies the Yang-Baxter equation and a cubic equation (α² - 1)(α + q²) = 0. This operator can be extended to a family of operators $h_j:=I_j\otimes \alpha \otimes I_{n-2-j}$ on ${\left(ℂ³\right)}^{\otimes n}$ with 0 ≤ j ≤ n - 2. These operators generate the cubic Hecke algebra ${\mathscr{H}}_{q,n}\left(2\right)$ associated with the quantum group $U_q\left(2\right)$. The purpose of this note is to present the construction.

In continuation of our earlier work [2] we describe the indecomposable representations and the Auslander-Reiten quivers of a family of vector space categories playing an important role in the study of domestic finite dimensional algebras over an algebraically closed field. The main results of the paper are applied in our paper [3] where we exhibit a wide class of almost sincere domestic simply connected algebras of arbitrary large finite global dimensions and describe their Auslander-Reiten quivers....

In this paper rings for which every $s$-torsion quasi-injective module is weakly $s$-divisible for a hereditary preradical $s$ are characterized in terms of the properties of the corresponding lattice of the (hereditary) preradicals. In case of a stable torsion theory these rings coincide with $TQI$-rings investigated by J. Ahsan and E. Enochs in [1]. Our aim was to generalize some results concerning $QI$-rings obtained by J.S. Golan and S.R. L’opez-Permouth in [12]. A characterization of the $QI$-property in the...

a recent paper of the first author and Kashyap, a new class of Banach modules over dual operator algebras is introduced. These generalize the W*-modules (that is, Hilbert C*-modules over a von Neumann algebra which satisfy an analogue of the Riesz representation theorem for Hilbert spaces), which in turn generalize Hilbert spaces. In the present paper, we describe these modules, giving some motivation, and we prove several new results about them.

If $X$ is a Tychonoff space, $C\left(X\right)$ its ring of real-valued continuous functions. In this paper, we study non-essential ideals in $C\left(X\right)$. Let $a$ be a infinite cardinal, then $X$ is called $a$-Kasch (resp. $\overline{a}$-Kasch) space if given any ideal (resp. $z$-ideal) $I$ with $gen\left(I\right)<a$ then $I$ is a non-essential ideal. We show that $X$ is an ${\aleph }_0$-Kasch space if and only if $X$ is an almost $P$-space and $X$ is an ${\aleph }_1$-Kasch space if and only if $X$ is a pseudocompact and almost $P$-space. Let $C_F\left(X\right)$ denote the socle of $C\left(X\right)$. For a topological space $X$ with only...

In 1964, Bertram Yood posed the following problem: whether the intersection of all closed maximal regular left ideals of a topological ring coincides with the intersection of all closed maximal regular right ideals of this ring. It is proved that these two intersections coincide for advertive and simplicial topological rings and, using this result, it is shown that the topological left radical and the topological right radical for every advertive and simplicial topological algebra coincide.

Given a pair M,M' of finite-dimensional modules over a domestic canonical algebra Λ, we give a fully verifiable criterion, in terms of a finite set of simple linear algebra invariants, deciding if M and M' lie in the same orbit in the module variety, or equivalently, if M and M' are isomorphic.

Let $R$ be a prime ring, with no non-zero nil right ideal, $d$ a non-zero drivation of $R$, $I$ a non-zero two-sided ideal of $R$. If, for any $x$, $y\in I$, there exists $n=n\left(x,y\right)\ge 1$ such that ${\left(d\left(\left[x,y\right]\right)-\left[x,y\right]\right)}^n=0$, then $R$ is commutative. As a consequence we extend the result to Lie ideals.

We study McCoy’s theorem to the skew Hurwitz series ring $(\mathrm{HR},\omega )$ for some different classes of rings such as: semiprime rings, APP rings and skew Hurwitz serieswise quasi-Armendariz rings. Moreover, we establish an equivalence relationship between a right zip ring and its skew Hurwitz series ring in case when a ring $R$ satisfies McCoy’s theorem of skew Hurwitz series.