Von Neumann Regular Research Articles

Ideally nil clean rings

A ring is called nil clean if every element can be written as the sum of an idempotent and a nilpotent. The class of nil clean rings has emerged as an important variant of the much-studied class of clean rings. However, the nil clean definition is fairly restrictive, and the collection of examples is rather constrained. In this paper, we propose an ideal-theoretic version of the nil clean concept, with the goal of creating a more expansive class of rings. Specifically, we define a ring to be ideally nil clean (INC) if every (two-sided) ideal can be written as the sum of an idempotent ideal and a nil ideal. By passing from elements to ideals, we vastly expand the collection of rings under consideration. For example, all artinian rings and all von Neumann regular rings are INC. We show that, in the commutative case, the INC rings are exactly the strongly π-regular rings. We also explore the relationship between the nil clean and INC properties. After establishing a robust collection of examples of INC rings, we explore the behavior of the INC condition under common ring extensions. In addition, we state several open questions and suggest areas for further investigation.

Ordered Field Valued Continuous Functions with Countable Range

For a Hausdorff zero-dimensional topological space X and a totally ordered field F with interval topology, let $$C_c(X,F)$$ be the ring of all F-valued continuous functions on X with countable range. It is proved that if F is either an uncountable field or countable subfield of $${\mathbb {R}}$$ , then the structure space of $$C_c(X,F)$$ is $$\beta _0X$$ , the Banaschewski Compactification of X. The ideals $$\{O^{p,F}_c:p\in \beta _0X\}$$ in $$C_c(X,F)$$ are introduced as modified countable analogue of the ideals $$\{O^p:p\in \beta X\}$$ in C(X). It is realized that $$C_c(X,F)\cap C_K(X,F)=\bigcap _{p\in \beta _0X{\setminus } X} O^{p,F}_c$$ , and this may be called a countable analogue of the well-known formula $$C_K(X)=\bigcap _{p\in \beta X{\setminus } X}O^p$$ in C(X). Furthermore, it is shown that the hypothesis $$C_c(X,F)$$ is a Von-Neumann regular ring is equivalent to amongst others the condition that X is a P-space.

A Characterization of von-Neumann Regular S-Acts

A Characterization of von-Neumann Regular S-Acts

A symmetric generalization of $$\pi $$-regular rings

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 $$\pi $$ -regular rings (in particular, of the von Neumann regular rings and of the strongly $$\pi $$ -regular rings). Some other close relationships with certain well-known classes of rings such as exchange rings, clean rings, nil-clean rings, etc., are also demonstrated. These results somewhat supply a recent publication of the author in Turk J Math (2019) as well as they somewhat expand the important role of the two examples of nil-clean rings obtained by Ster in Linear Algebra Appl (2018). Likewise, the obtained symmetrization supports that similar property for exchange rings established by Khurana et al. in Algebras Represent Theory (2015).

A study of non-commutative Von-Neumann regular rings

In this paper, we study the Von-Neumann regular elements of non-commutative rings M2(ℤ2) and M2(ℤ3). We prove that M2(ℤ2) and M3(ℤ3) are Von-Neumann rings. We also give code in C++to compute Von-Neumann regular elements of M2(ℤn).

Weak global dimension and CM-freeness of rings

Let $R$ be a ring with finite weak global dimension. We prove that $R$ is (strongly) CM-free. As applications,we obtain some triangulated equivalences between some compactly-generated homotopy categories and relative derived categories.We classify the definable categories of such equivalent triangulated categories when the weak global dimension of $R$ is at most 1, and show that such equivalent triangulated categoriessatisfy Telescope conjecture when $R$ is a von Neumann regular ring. We also improve the condition such that a triangulated equivalence about the extended Grothendieck duality holdsfrom the finiteness of the left and right global dimension of $R$ to only the finiteness of the right global dimension of $R$.

English

In Theorem 6 of [1], if R is a von Neumann regular ring, then every 2-prime ideal of R is a prime ideal. But the converse of this implication does not hold. Thus, we correct Theorem 6 of [1] as follows:

Self-injective von Neumann regular rings and Köthe's conjecture

One of the many equivalent formulation of the Köthe's conjecture is the assertion that there exists no unital ring which contains two nil right ideals whose sum is not nil. We discuss several consequences of an observation that if the Koethe conjecture fails then there exists a counterexample in the form of a countable local subring of a suitable self-injective prime von Neumann regular unital ring.

An alternative perspective on pure-projectivity of modules

The study of pure-projectivity is accessed from an alternative point of view. Given modules M and N, M is said to be N-pure-subprojective if for every pure epimorphism $$g: B\rightarrow N$$ and homomorphism $$f : M \rightarrow N$$ , there exists a homomorphism $$h: M\rightarrow B$$ such that $$gh=f$$ . For a module M, the pure-subprojectivity domain of M is defined to be the collection of all modules N such that M is N-pure-subprojective. We obtain characterizations for various types of rings and modules, including FP-injective and FP-projective modules, von Neumann regular rings and pure-semisimple rings in terms of pure-subprojectivity domains. As pure-subprojectivity domains clearly include all pure-projective modules, a reasonable opposite to pure-projectivity in this context is obtained by considering modules whose pure-subprojectivity domain consists of only pure-projective. We refer to these modules as psp-poor. Properties of pure-subprojectivity domains and of psp-poor modules are studied.

A characterization of graded von Neumann regular rings with applications to Leavitt path algebras

We prove a new characterization of graded von Neumann regular rings involving the recently introduced class of nearly epsilon-strongly graded rings. As our main application, we generalize Hazrat's result that Leavitt path algebras over fields are graded von Neumann regular. More precisely, we show that a Leavitt path algebra LR(E) with coefficients in a unital ring R is graded von Neumann regular if and only if R is von Neumann regular. We also prove that both Leavitt path algebras and corner skew Laurent polynomial rings over von Neumann regular rings are semiprimitive and semiprime. Thereby, we generalize a result by Abrams and Aranda Pino on the semiprimitivity of Leavitt path algebras over fields.

A note on r-precious ring

An element of a ring [Formula: see text] is said to be [Formula: see text]-precious if it can be written as the sum of a von Neumann regular element, an idempotent element and a nilpotent element. If all the elements of a ring [Formula: see text] are [Formula: see text]-precious, then [Formula: see text] is called an [Formula: see text]-precious ring. We study some basic properties of [Formula: see text]-precious rings. We also characterize von Neumann regular elements in [Formula: see text] when [Formula: see text] is a Euclidean domain and by this argument, we produce elements that are [Formula: see text]-precious but either not [Formula: see text]-clean or not precious.

Coloring of cozero-divisor graphs of commutative von Neumann regular rings

Let R be a commutative ring with non-zero identity. The cozero-divisor graph of R, denoted by $$\Gamma ^{\prime }(R)$$ , is a graph with vertices in $$W^*(R)$$ , which is the set of all non-zero and non-unit elements of R, and two distinct vertices a and b in $$W^*(R)$$ are adjacent if and only if $$a\not \in Rb$$ and $$b\not \in Ra$$ . In this paper, we show that the cozero-divisor graph of a von Neumann regular ring with finite clique number is not only weakly perfect but also perfect. Also, an explicit formula for the clique number is given.

Generalization of $z$-ideals in right duo rings

The aim of this paper is to generalize the notion of $z$-ideals to arbitrary noncommutative rings. A left (right) ideal $I$ of a ring $R$ is called a left (right) $z$-ideal if $M_a \subseteq I$, for each $a\in I$, where $M_a$ is the intersection of all maximal ideals containing $a$. For every two left ideals $I$ and $J$ of a ring $R$, we call $I$ a left $z_J$-ideal if $M_a \cap J \subseteq I$, for every $a\in I$, whenever $ J \nsubseteq I$ and $I$ is a $z_J$-ideal, we say that $I$ is a left relative $z$-ideal. We characterize the structure of them in right duo rings. It is proved that a duo ring $R$ is von Neumann regular ring if and only if every ideal of $R$ is a $z$-ideal. Also, every one sided ideal of a semisimple right duo ring is a $z$-ideal. We have shown that if $I$ is a left $z_J$-ideal of a $p$-right duo ring, then every minimal prime ideal of $I$ is a left $z_J$-ideal. Moreover, if every proper ideal of a $p$-right duo ring $R$ is a left relative $z$-ideal,then every ideal of $R$ is a $z$-ideal.

On Singly Flat and Singly Injective Modules

Recall that a left module A (resp. right module B) is said to be singly injective (resp. singly flat) if $$Ext^{1}_{R}(F/K, A) = 0$$ (resp. $${\text {Tor}}_{1}^{R}(B, F/K)= 0$$ ) for any cyclic submodule K of any finitely generated free left R-module F. In this paper, we continue to study and investigate the homological objects related to singly flat and singly injective modules and module homomorphisms. Along the way, the right orthogonal class of singly flat right modules and the left orthogonal class of singly injective left modules are introduced and studied. These concepts are used to extend the some known results and to characterize pseudo-coherent rings and left singly injective rings. In terms of some derived functors, some homological dimensions are investigated. As applications, some new characterizations of von Neumann regular rings and left PP rings are given. Finally, we study the singly flatness and singly injectivity of homomorphism modules over a commutative ring.

On graded primitive Leavitt path algebras

Graded primitive Leavitt path algebras of arbitrary directed graphs over a field [Formula: see text] are completely characterized by means of graphical conditions. Necessary and sufficient conditions are given under which a graded prime Leavitt path algebra becomes graded primitive and this leads to answering the graded version of a question of Kaplansky on von Neumann regular prime rings in the context of Leavitt path algebras.

ON VON NEUMANN REGULAR MODULES

Adv. Math. Sci. J. Vol. 9, No. 4 Open Access ON VON NEUMANN REGULAR MODULES G. N. Sudharshana and D. Siva Kumar DOI: https://doi.org/10.37418/amsj.9.4.51 Full Text Abstract Abstract Results gotten for a module *** QuickLaTeX cannot compile formula: M *** Error message: Cannot connect to QuickLaTeX server: cURL error 60: SSL certificate problem: certificate has expired Please make sure your server/PHP settings allow HTTP requests to external resources ("allow_url_fopen", etc.) These links might help in finding solution: http://wordpress.org/extend/plugins/core-control/ http://wordpress.org/support/topic/an-unexpected-http-error-occurred-during-the-api-request-on-wordpress-3?replies=37 over a commutative ring have … page AMSJ9-N4-51 Read More »

Endoregular modules

We study modules whose endomorphism rings possess various forms of von Neumann regularity. We characterize these “regularity” properties for several classes of modules, including completely decomposable modules and finitely presented modules over commutative rings. We generalize many of the classic results about abelian groups with regular endomorphism rings to modules over one-dimensional commutative rings with Noetherian spectrum. To facilitate this study, we define a module M over a commutative ring R to be weakly endoregular if xM and (0:M(x)) are direct summands of M for each x∈R. We give several characterizations of weak endoregularity for various classes of modules.

New results on common properties of the products AC and BA, II

AbstractIn this note, we continue to investigate common properties of the products in the setting of rings, bounded linear operators, or Banach algebras. We prove: (i) If are elements in a unital associative ring satisfying , then von Neumann regularity (resp. generalized Fredholmness relative to an ideal of ) of is converted into that of . (ii) If are bounded linear operators satisfying , then and share common complementability of kernels and ranges. (iii) If are elements in a unital semisimple Banach algebra satisfying and is a trace ideal of such that soc kh(soc, then and share common Fredholmness relative to and have the same abstract index. A similar result holds for B‐Fredholmness in primitive Banach algebra.

The realization problem for finitely generated refinement monoids

We show that every finitely generated conical refinement monoid can be represented as the monoid V(R) of isomorphism classes of finitely generated projective modules over a von Neumann regular ring R. To this end, we use the representation of these monoids provided by adaptable separated graphs. Given an adaptable separated graph (E, C) and a field K, we build a von Neumann regular K-algebra QK(E,C) and show that there is a natural isomorphism between the separated graph monoid M(E, C) and the monoid V(QK(E,C)).

Reverse decomposition of unipotents over noncommutative rings I: General linear groups

Recently it has been proved that if σ∈GLn(R) where R is a commutative ring and n≥3, then each of the matrices tkl(σij)(i≠j,k≠l) is a product of eight En(R)-conjugates of σ and σ−1. In this article we show that similar results hold true if R is a (noncommutative) von Neumann regular ring, or a Banach algebra, or a ring satisfying a stable range condition, or a ring with Euclidean algorithm, or an almost commutative ring.