Related papers: Direct products of modules and the pure semisimpli…

We exhibit in this article a contraction of the direct product Lie algebra $g\oplus g$ of a finite-dimensional complex Lie algebra $g$ onto the semi-direct product Lie algebra $g\rtimes g$, where the first factor $g$ is viewed as a trivial…

If a left-product $M_n... M_1$ of square complex matrices converges to a nonnull limit when $n\to\infty$ and if the $M_n$ belong to a finite set, it is clear that there exists an integer $n_0$ such that the $M_n$, $n\ge n_0$, have a common…

Auslander and Ringel-Tachikawa have shown that for an artinian ring R of finite representation type, every R-module is the direct sum of finitely generated indecomposable R-modules. In this paper, we will adapt this result to finite…

We study the structure of the indecomposable direct summands of tensor products of two restricted simple $SL_3(K)$-modules, where $K$ is an algebraically closed field of characteristic $p \geq 5$. We give a characteristic-free algorithm for…

We study Artin algebras $A$ and commutative Noetherian complete local rings $R$ in connection with the following decomposition property of Gorenstein-projective modules: $(*)$ any Gorenstein-projective module is a direct sum of finitely…

In this paper we prove two new results about closed semigroups in the family of solvable groups H_{mn} that are semidirect products of R^m and R^n, and for which the structure homomorphism maps nontrivially into the center of Aut(R^n). The…

Monogeny classes and epigeny classes have proved to be useful in the study of direct sums of uniserial modules and other classes of modules. In this paper, we show that they also turn out to be useful in the study of direct products.

We consider the class of crossed products of noetherian domains with universal enveloping algebras of Lie algebras. For algebras from this class we give a sufficient condition for the existence of projective non-free modules. This class…

We study surjective homomorphisms f:\prod_I A_i\to B of not-necessarily-associative algebras over a commutative ring k, for I a generally infinite set; especially when k is a field and B is countable-dimensional over k. Our results have the…

Matlis duality for modules over commutative rings gives rise to the notion of Matlis reflexivity. It is shown that Matlis reflexive modules form a Krull-Schmidt category. For noetherian rings the absence of infinite direct sums is a…

An artin algebra $A$ is said to be CM-finite if there are only finitely many, up to isomorphisms, indecomposable finitely generated Gorenstein-projective $A$-modules. We prove that for a Gorenstein artin algebra, it is CM-finite if and only…

Let $G$ be an abelian group of order $n$ and let $R$ be a commutative ring which admits a homomorphism ${\Bbb Z}[\zeta_{n}]\ra R$, where $\zeta_{n}$ is a (complex) primitive $n$-th root of unity. Given a finite $R[G\e]$-module $M$, we…

We prove that the tensor product of a simple and a finite dimensional $\mathfrak{sl}_n$-module has finite type socle. This is applied to reduce classification of simple $\mathfrak{q}(n)$-supermodules to that of simple…

The usual nonnegative modulus function is based on addition. A natural different modulus function on the set of positive reals is introduced. Arguments for results for series through the usual modulus function are transformed to arguments…

Let k be an infinite field, I an infinite set, V a k-vector-space, and g:k^I\to V a k-linear map. It is shown that if dim_k(V) is not too large (under various hypotheses on card(k) and card(I), if it is finite, respectively countable,…

We investigate the almost Cohen-Macaulay property and the Serre-type condition $(C_n),\ n\in\mathbb{N},$ for noetherian algebras and modules. More precisely, we find permanence properties of these conditions with respect to tensor products…

Let $Q$ be a Noetherian ring with finite Krull dimension and let $\mathbf{f}= f_1,... f_c$ be a regular sequence in $Q$. Set $A = Q/(\mathbf{f})$. Let $I$ be an ideal in $A$, and let $M$ be a finitely generated $A$-module with $\projdim_Q…

An algebra $A$ is said to be directly finite if each left invertible element in the (conditional) unitization of $A$ is right invertible. We show that the reduced group ${\rm C}^\ast$-algebra of a unimodular group is directly finite,…

Tensor product of irreducible modules of highest weight over a semi-simple quantum group is semi-simple if and only if a natural contravariant form is non-degenerate when restricted to the span of singular vectors. We express this…

The direct product $\mathbb{N}\times\mathbb{N}$ of two free monogenic semigroups contains uncountably many pairwise non-isomorphic subdirect products. Furthermore, the following hold for $\mathbb{N}\times S$, where $S$ is a finite…