Related papers: On our paper `Almost Free Splitter', a correction

Suppose $X$ is a smooth affine real variety and $\mathscr{E}$ is a vector bundle over $X$. We analyze the problem of splitting off a free rank one summand from $\mathscr{E}$ in corank $0$ and $1$. The problem in corank $0$ can be viewed as…

In "Almost Free Modules, Set-theoretic Methods", Eklof and Mekler raised the question about the existence of dual abelian groups G which are not isomorphic to Z+G. Recall that G is a dual group if G ~ D^* for some group D with D^*=Hom(D,Z).…

A set of elements of a finite abelian group is called sum-free if it contains no Schur triple, i.e., no triple of elements $x,y,z$ with $x+y=z$. The study of how large the largest sum-free subset of a given abelian group is had started more…

A $G$-grading on an algebra is called multiplicity free if each homogeneous component of the grading is 1-dimensional, where $G$ is an abelian group. We introduce skew root systems of Lie type and skew root systems of Jordan type…

Recall that a homomorphism of $R$-modules $\pi: G\to H$ is called a {\it cellular cover} over $H$ if $\pi$ induces an isomorphism $\pi_*: \Hom_R(G,G)\cong \Hom_R(G,H),$ where $\pi_*(\varphi)= \pi \varphi$ for each $\varphi \in \Hom_R(G,G)$…

Let $A$ be a quasi-hereditary algebra. We prove that in many cases, a tilting module is rigid (i.e. has identical radical and socle series) if it does not have certain subquotients whose composition factors extend more than one layer in the…

The normal form theorem, proved in R. Laver, On the left distributive law and the freeness of an algebra of elementary embeddings, Advances in Mathematics 91 (1992), 209-231, for the free algebra $\Cal A$ on one generator $x$ satisfying the…

Let $S$ be a deeply embedded, equicharacteristic, Artinian Gorenstein local ring. We prove that if $R$ is a non-Gorenstein quotient of $S$ of small colength, then every totally reflexive $R$-module is free. Indeed, the second syzygy of the…

Let $(R, \m)$ be a commutative Noetherian local ring with $\m^3 =(0)$. We give a condition for $R$ to have a non-free module of G-dimension zero. We shall also construct a family of non-isomorphic indecomposable modules of G-dimension zero…

Let $G$ be an algebraic group, $X$ a generically free $G$-variety, and $K=k(X)^G$. A field extension $L$ of $K$ is called a splitting field of $X$ if the image of the class of $X$ under the natural map $H^1(K, G) \mapsto H^1(L, G)$ is…

A ring is clean (almost clean) if each of its elements is the sum of a unit (regular element) and an idempotent. A module is clean (almost clean) if its endomorphism ring is clean (almost clean). We show that every quasi-continuous and…

For the cyclic group $C_2$ we give a complete description of the derived category of perfect complexes of modules over the constant Mackey ring $\underline{\mathbb{Z}/\ell}$, for $\ell$ a prime. This is fairly simple for $\ell$ odd, but for…

Let $R$ be a Noetherian commutative ring and $M$ an $R$-module with $\operatorname{pd_R} M\le 1$ that has rank. Necessary and sufficient conditions were provided by Lebelt for an exterior power $\wedge^k M$ to be torsion free. When $M$ is…

We prove a version of Shelah's Categoricity Conjecture for arbitrary deconstructible classes of modules. Moreover, we show that if $\mathcal{A}$ is a deconstructible class of modules that fits in an abstract elementary class…

The purpose of this paper and its sequel, is to introduce a new class of modules over a commutative ring $R$, called $\mathbb{P}$-radical modules (modules $M$ satisfying the prime radical condition "$(\sqrt[p]{{\cal{P}}M}:M)={\cal{P}}$" for…

We extend the concept of "almost indiscernible theory" introduced by Pillay and Sklinos in [Bull. Symb. Log., 2015] (which was itself a modernization and expansion of Baldwin and Shelah [Algebra Universalis, 1983]), to uncountable languages…

We show that there exists a stably free module over a polynomial ring which is not extended from the ground ring. This provides a counterexample to the Hermite ring conjecture.

We prove that stably free modules of rank d-1 over a smooth affine algebra of dimension d over an algebraically closed field k are free, provided (d-1)! is nonzero in k.

We show that, over any ring, a module $C$ is a codomain of a right almost split map if and only if $C$ is a finitely presented module with local endomorphism ring; thus we give an answer to a 40 years old question by M. Auslander. Using the…

Given a partially ordered set $P$ there exists the most general Boolean algebra $F(P)$ which contains $P$ as a generating set, called the {\it free Boolean algebra} over $P$. We study free Boolean algebras over posets of the form $P=P_0\cup…