Related papers: Von Neumann Regular $\mathcal{C}^{\infty}-$Rings a…
In this paper, we introduce the concept of a "von Neumann regular $\mathcal{C}^{\infty}$-ring", which is a model for a specific equational theory. We delve into the characteristics of these rings and demonstrate that each Boolean space can…
We use the concept of a regular object with respect to another object in an arbitrary category, defined in \cite{dntd}, in order to obtain the transfer of regularity in the sense of Zelmanowitz between the categories $R-$mod and $S-$mod,…
In this paper we present classifying toposes for the following theories: the theory of $\mathcal{C}^{\infty}-$rings, the theory of local $\mathcal{C}^{\infty}-$rings and the theory of von Neumann regular $\mathcal{C}^{\infty}-$rings. The…
In this paper we present some basic results of the Universal Algebra of $\mathcal{C}^\infty$-rings which were nowhere to be found in the current literature. The outstanding book of I. Moerdijk and G. Reyes,[24], presents the basic (and…
Different and distinct notions of regularity for modules exist in the literature. When these notions are restricted to commutative rings, they all coincide with the well-known von-Neumann regularity for rings. We give new characterizations…
In this article, we give a complete characterization of semigroup graded rings which are graded von Neumann regular. We also demonstrate our results by applying them to several classes of examples, including matrix rings and groupoid graded…
Separative von Neumann regular rings exist in abundance. For example, all regular self-injective rings, unit regular rings, regular rings with a polynomial identity are separative. It remains open whether there exists a non-separative…
In this dissertation we study the category of completely positive normal contractive maps between von Neumann algebras. It includes an extensive introduction to the basic theory of $C^*$-algebras and von Neumann algebras.
We give a complete description of ring isomorphisms between algebras of measurable operators affiliated with von Neumann algebras of type II$_1.$
We show that every ring isomorphism between the algebras of locally measurable operators for type II$_\infty$ von Neumann algebras is similar to a real $^*$-isomorphism. This together with previous results by the author and…
We introduce and investigate the so-called D-regularly nil clean rings by showing that these rings are, in fact, a non-trivial generalization of the classical von Neumann regular rings and of the strongly $\pi$-regular rings. Some other…
The analytic von Neumann regular closure $R(\Gamma)$ of a complex group algebra $\C\Gamma$ was introduced by Linnell and Schick. This ring is the smallest $*$-regular subring in the algebra of affiliated operators $U(\Gamma)$ containing…
The purpose of this work is to investigate various notions of regularity from the perspective of finiteness conditions, with the ultimate goal of identifying broad classes of rings that are $\mathsf{K}_0$-regular. In this direction, we…
The present paper is devoted to study of ring isomorphisms of $\ast$-subalgebras of Murray--von Neumann factors. Let $\cM,$ $\cN$ be von Neumann factors of type II$_1,$ and let $S(\cM),$ $S(\cN)$ be the $\ast$-algebras of all measurable…
This paper develops a theory of $C^\infty$-superrings and their associated $C^\infty$-superschemes. We prove a key equivalence between the category of fair affine $C^\infty$-superschemes and the category of fair $C^\infty$-superrings. We…
A ring R is said to be VNL if for any a in R, either a or 1-a is (von Neumann) regular. The class of VNL rings lies properly between the exchange rings and (von Neumann) regular rings. We characterize abelian VNL rings. We also characterize…
The concepts of Boolean metric space and convex combination are used to characterize polynomial maps in a class of commutative Von Neumann regular rings including Boolean rings and p-rings, that we have called CFG-rings. In those rings, the…
We present, in the same vein as in [20] and [21], some results of the so-called "Smooth (or $\mathcal{C}^\infty$) Commutative Algebra", a version of Commutative Algebra of $\mathcal{C}^{\infty}-$rings instead of ordinary commutative unital…
In the present work we carry on the study of the order theory for ($\mathcal{C}^{\infty}-$-reduced) $\mathcal{C}^{\infty}-$-rings initiated in \cite{rings1} (see also \cite{BM2}). In particular, we apply some results of the order theory of…
In the given article it is introduced new notions of a C$^*$-algebra of von Neumann type I and C$^*$-algebras of types I$_n$, II, II$_1$, II$_\infty$ and III. It is proved that any GCR-algebra is a C$^*$-algebra of von Neumann type I, and a…