Related papers: The realization problem for some wild monoids and …

We survey recent progress on the realization problem for von Neumann regular rings, which asks whether every countable conical refinement monoid can be realized as the monoid of isoclasses of finitely generated projective right $R$-modules…

We show that every finitely generated conical refinement monoid can be represented as the monoid $\mathcal V(R)$ of isomorphism classes of finitely generated projective modules over a von Neumann regular ring $R$. To this end, we use the…

We define a subclass of separated graphs, the class of adaptable separated graphs, and study their associated monoids. We show that these monoids are primely generated conical refinement monoids, and we explicitly determine their associated…

Graph monoids arise naturally in the study of non-stable K-theory of graph C*-algebras and Leavitt path algebras. They play also an important role in the current approaches to the realization problem for von Neumann regular rings. In this…

For every infinite cardinal number $\kappa$, $\kappa$-monoids and their realization have recently been introduced and studied by Nazemian and Smertnig. A $\kappa$-monoid $H$ has a realization to a ring $R$ if there exists an element $x \in…

Let $K$ be a field. We attach to each finite poset $\mathbb P$ a von Neumann regular $K$-algebra $Q_K(\mathbb P)$ in a functorial way. We show that the monoid of isomorphism classes of finitely generated projective $Q_K(\mathbb P)$-modules…

The class of refinement monoids (abelian monoids satisfying the Riesz refinement property) is subdivided into those which are tame, defined as being an inductive limit of finitely generated refinement monoids, and those which are wild,…

The smooth (resp. metric and complex) Nielsen Realization Problem for K3 surfaces $M$ asks: when can a finite group $G$ of mapping classes of $M$ be realized by a finite group of diffeomorphisms (resp. isometries of a Ricci-flat metric, or…

Let G be a group and let K be a field of characteristic zero. We shall prove that KG can be embedded into a von Neumann unit-regular ring. In the course of the proof, we shall obtain a result relevant to the Atiyah conjecture.

Let $S$ be a submonoid of a free Abelian group of finite rank. We show that if $k$ is a field of prime characteristic such that the monoid $k$-algebra $k[S]$ is split $F$-regular, then $k[S]$ is a finitely generated $k$-algebra, or…

We consider realization and isomorphism problems for formal matrix rings over a given ring. Principal multiplier matrices of such rings play an important role in this case.\\ The work of A.A.Tuganbaev is supported by Russian Scientific…

We characterize, in terms of elementary properties, the abelian monoids which are direct limits of finite direct sums of monoids of the form $(Z/nZ)\sqcup\{0\}$ (where 0 is a new zero element), for positive integers $n$. The key properties…

We demonstrate that the ring of invariants for the natural action of a subgroup G of GL_n(F_q) on a polynomial ring R=K[X_1,...,X_n] need not be F-pure. In these examples G is the symplectic group over a finite field, and the invariant…

A half a century ago, George Bergman introduced stunning machinery which would realise any commutative conical monoid as the non-stable $K$-theory of a ring. The ring constructed is ``minimal" or ``universal". Given the success of graded…

We raise the question of the realizability of permutation modules in the context of Kahn's realizability problem for abstract groups and the $G$-Moore space problem. Specifically, given a finite group $G$, we consider a collection…

Idealization of a module $K$ over a commutative ring $S$ produces a ring having $K$ as an ideal, all of whose elements are nilpotent. We develop a method that under suitable field-theoretic conditions produces from an $S$-module $K$ and…

The realization problem asks which algebras can be realized as the cohomology of spaces. We study this problem in the context of the orders in a graded rational exterior algebra on three generators. An order is a subring whose underlying…

Kuratowski's closure-complement problem gives rise to a monoid generated by the closure and complement operations. Consideration of this monoid yielded an interesting classification of topological spaces, and subsequent decades saw further…

Let M be an irreducible normal algebraic monoid with unit group G. It is known that G admits a Rosenlicht decomposition, G=G_antG_aff, where G_ant is the maximal anti-affine subgroup of G, and G_aff the maximal normal connected affine…

Complexity problems associated with finite rings and finite semigroups, particularly semigroups of matrices over a field and the Rees matrix semigroups, are examined. Let M_nF be the ring of n x n matrices over the finite field F and let…