We generalize the results of Kahn about a correspondence between Cohen-Macaulay modules and vector bundles to non-commutative surface singularities. As an application, we give examples of non-commutative surface singularities which are not Cohen-Macaulay finite, but are Cohen-Macaulay tame.

Suppose that A and B are unital Banach algebras with units $1_A$ and $1_B$, respectively, M is a unital Banach A,B-module, $=\left[\begin{array}{cc}A& M\\ 0& B\end{array}\right]$ is the triangular Banach algebra, X is a unital -bimodule, $X_{AA}=1_{A}X{1}_A$, $X_{BB}=1_{B}X{1}_B$, $X_{AB}=1_{A}X{1}_B$ and $X_{BA}=1_{B}X{1}_A$. Applying two nice long exact sequences related to A, B, , X, $X_{AA}$, $X_{BB}$, $X_{AB}$ and $X_{BA}$ we establish some results on (co)homology of triangular Banach algebras.

Let $G$ be an abelian group, $R$ a commutative ring of prime characteristic $p$ with identity and $R_{t}G$ a commutative twisted group ring of $G$ over $R$. Suppose $p$ is a fixed prime, $G_p$ and $S\left(R_{t}G\right)$ are the $p$-components of $G$ and of the unit group $U\left(R_{t}G\right)$ of $R_{t}G$, respectively. Let $R^{*}$ be the multiplicative group of $R$ and let $f_{\alpha }\left(S\right)$ be the $\alpha$-th Ulm-Kaplansky invariant of $S\left(R_{t}G\right)$ where $\alpha$ is any ordinal. In the paper the invariants $f_n\left(S\right)$, $n\in \mathbb{N}\cup \left\{0\right\}$, are calculated, provided $G_p=1$. Further, a commutative ring $R$ with identity of prime characteristic $p$ is said...

Two new results concerning complements in a semisimple Hopf algebra are proved. They extend some well-known results from group theory. The uniqueness of a Krull-Schmidt-Remak type decomposition is proved for semisimple completely reducible Hopf algebras.

Trivial extensions of a certain subclass of minimal 2-fundamental algebras are examined. For such algebras the characterization of components of the Auslander-Reiten quiver which contain indecomposable projective modules is given.

Let $A=K[a_1,...,a_n]$ be a (noncommutative) solvable polynomial algebra over a field $K$ in the sense of A. Kandri-Rody and V. Weispfenning [Non-commutative Gröbner bases in algebras of solvable type, J. Symbolic Comput. 9 (1990), 1–26]. This paper presents a comprehensive study on the computation of minimal free resolutions of modules over $A$ in the following two cases: (1) $A={⨁}_{p\in \mathbb{N}}{A}_p$ is an $\mathbb{N}$-graded algebra with the degree-0 homogeneous part $A_0=K$; (2) $A$ is an $\mathbb{N}$-filtered algebra with the filtration ${\left\{F_{p}A\right\}}_{p\in \mathbb{N}}$ determined by a positive-degree...