Von Neumann Regular Research Articles

The Groupoids of Adaptable Separated Graphs and Their Type Semigroups

AbstractGiven an adaptable separated graph, we construct an associated groupoid and explore its type semigroup. Specifically, we first attach to each adaptable separated graph a corresponding semigroup, which we prove is an $E^*$-unitary inverse semigroup. As a consequence, the tight groupoid of this semigroup is a Hausdorff étale groupoid. We show that this groupoid is always amenable and that the type semigroups of groupoids obtained from adaptable separated graphs in this way include all finitely generated conical refinement monoids. The first three named authors will utilize this construction in forthcoming work to solve the realization problem for von Neumann regular rings, in the finitely generated case.

Mod-retractable strongly π-regular rings and group algebras

In this work, we show that a two-sided mod-retractable and strongly π-regular ring is two-sided semiartinian. On the other hand, a strongly π-regular von Neumann regular ring is left mod-retractable if and only if it is a left semiartinian V-ring. We apply these results to group algebras and thereby we give the complete structure of groups whose group algebras are mod-retractable.Communicated by Toma Albu

WHEN THE HEWITT REALCOMPACTIFICATION AND THE P-COREFLECTION COMMUTE

If X is a Tychonoff space then its P -coreflection Xδ is a Tychonoff space that is a dense subspace of the realcompact space (υX)δ, where υX denotes the Hewitt realcompactification of X. We investigate under what conditions Xδ is C-embedded in (υX)δ, i.e. under what conditions υ(Xδ )=( υX)δ. An example shows that this can fail for the product of a compact space and a P -space. It is possible for a von Neumann regular ring A to be isomorphic to a C(Y ) and lie between C(X) and C(Xδ) without being isomorphic to C(Xδ). This cannot occur if X is realcompact or more generally if υ(Xδ )=( υX)δ. Applications are given to the epimorphic hull of C(X).

Invariance and parallel sums

In this paper, the notions of invariance and parallel sums as defined by Anderson and Duffin for matrices [Series and parallel addition of matrices, J. Math. Anal. Appl. 26 (1969) 576–594] are generalized to von Neumann regular rings.

Another multiplicative property of a regular ring that has no multiplicative proof

We explicitly record the largely unnoticed property of a von Neumann regular ring R that every element a of R has a quasi-inverse $$a^{\prime }$$ such that $$aa^{\prime }$$ and $$a^{\prime }a$$ commute. We also show that this property can fail in a regular semigroup S, whence no purely multiplicative proof is available in regular rings.

Radical factorization for trivial extensions and amalgamated duplication rings

Let [Formula: see text] be a commutative ring extension such that [Formula: see text] is a trivial extension of [Formula: see text] (denoted by [Formula: see text]) or an amalgamated duplication of [Formula: see text] along some ideal of [Formula: see text] (denoted by [Formula: see text]. This paper examines the transfer of AM-ring, N-ring, SSP-ring and SP-ring between [Formula: see text] and [Formula: see text]. We study the transfer of those properties to trivial ring extension. Call a special SSP-ring an SSP-ring of the following type: it is the trivial extension of [Formula: see text] by a C-module [Formula: see text], where [Formula: see text] is an SSP-ring, [Formula: see text] a von Neumann regular ring and [Formula: see text] a multiplication C-module. We show that every SSP-ring with finitely many minimal primes which is a trivial extension is in fact special. Furthermore, we study the transfer of the above properties to amalgamated duplication along an ideal with some extra hypothesis. Our results allows us to construct nontrivial and original examples of rings satisfying the above properties.

The Dual Baer Criterion for non-perfect rings

Abstract Baer’s Criterion for Injectivity is a useful tool of the theory of modules. Its dual version (DBC) is known to hold for all right perfect rings, but its validity for the non-right perfect ones is a complex problem (first formulated by C. Faith [Algebra. II. Ring Theory, Springer, Berlin, 1976]). Recently, it has turned out that there are two classes of non-right perfect rings: (1) those for which DBC fails in ZFC, and (2) those for which DBC is independent of ZFC. First examples of rings in the latter class were constructed in [J. Trlifaj, Faith’s problem on R-projectivity is undecidable, Proc. Amer. Math. Soc. 147 2019, 2, 497–504]; here, we show that this class contains all small semiartinian von Neumann regular rings with primitive factors artinian.

Refinement monoids and adaptable separated graphs

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 I-systems. We also show that any finitely generated conical refinement monoid can be represented as the monoid of an adaptable separated graph. These results provide the first step toward an affirmative answer to the Realization Problem for von Neumann regular rings, in the finitely generated case.

Some algebraic and homological properties of a family of quotients of the Rees algebra

Let [Formula: see text] be a commutative ring and let [Formula: see text] be a proper ideal of [Formula: see text]. In this paper, we study some algebraic and homological properties of a family of rings [Formula: see text], with [Formula: see text], that are obtained as quotients of the Rees algebra associated with the ring [Formula: see text] and the ideal [Formula: see text]. Specially, we study when [Formula: see text] is a von Neumann regular ring, a semisimple ring and a Gaussian ring. Also, we study the classical global and weak global dimensions of [Formula: see text]. Finally, we investigate some homological properties of [Formula: see text]-modules and we show that [Formula: see text] and [Formula: see text] are Gorenstein projective [Formula: see text]-modules, provided some special conditions.

When is every module with essential socle a direct sum of automorphism-invariant modules?

The purpose of this paper is to study the structure of rings over which every essential extension of a direct sum of a family of simple modules is a direct sum of automorphism-invariant modules. We show that if [Formula: see text] is a right quotient finite dimensional (q.f.d.) ring satisfying this property, then [Formula: see text] is right Noetherian. Also, we show a von Neumann regular (semiregular) ring [Formula: see text] with this property is Noetherian. Moreover, we prove that a commutative ring with this property is an Artinian principal ideal ring.

Regular elements determined by generalized inverses

In a semiprime ring, von Neumann regular elements are determined by their inner inverses. In particular, for elements [Formula: see text] of a von Neumann regular ring [Formula: see text], [Formula: see text] if and only if [Formula: see text], where [Formula: see text] denotes the set of inner inverses of [Formula: see text]. We also prove that, in a semiprime ring, the same is true for reflexive inverses.

Leavitt Path Algebras of Hypergraphs

We define Leavitt path algebras of hypergraphs generalizing simultaneously Leavitt path algebras of separated graphs and Leavitt path algebras of vertex-weighted graphs (i.e. weighted graphs that have the property that any two edges emitted by the same vertex have the same weight). We investigate the Leavitt path algebras of hypergraphs in terms of linear bases, Gelfand–Kirillov dimension, ring-theoretic properties (e.g. simplicity, von Neumann regularity and Noetherianess), K-theory and graded K-theory. By doing so we obtain new results on the Gelfand–Kirillov dimension and graded K-theory of Leavitt path algebras of separated graphs and on the graded K-theory of Leavitt path algebras of vertex-weighted graphs.

Rings in which every left zero-divisor is also a right zero-divisor and conversely

A ring [Formula: see text] is called eversible if every left zero-divisor in [Formula: see text] is also a right zero-divisor and conversely. This class of rings is a natural generalization of reversible rings. It is shown that every eversible ring is directly finite, and a von Neumann regular ring is directly finite if and only if it is eversible. We give several examples of some important classes of rings (such as local, abelian) that are not eversible. We prove that [Formula: see text] is eversible if and only if its upper triangular matrix ring [Formula: see text] is eversible, and if [Formula: see text] is eversible then [Formula: see text] is eversible.

Rings described by some special morphisms of modules

The category of morphisms of R-modules is an expansion of the category of R-modules. Using some special morphisms of modules such as phantom, -phantom, flat and absolutely pure morphisms, we give some new characterizations of left coherent rings, von Neumann regular rings, left semihereditary rings, left Noetherian rings, and right perfect rings.

On clean and regular elements of noncommutative ring extensions

Let R be an associative ring with identity and α be an endomorphism of R. In this article, we are interested to study the some of relations between a ring R and that of D. A. Jordan’s construction of the ring A(R,α) as well as the skew Laurent polynomial ring . The main propose of this article is to characterize the unit elements, the idempotent elements, von Neumann regular elements, π-regular elements, von Neumann local elements and also the clean elements of the skew Laurent polynomial ring as well as Jordan’s construction of the ring A(R,α). Applying these characterizations, one can easily get some nice radical-theoretic properties of the mentioned classes of rings.

Direct finiteness of representable regular $$*$$ ∗ -rings

We show that a von Neumann regular ring with involution is directly finite provided that it admits a representation as a $$*$$ -ring of endomorphisms of a vector space endowed with a non-degenerate orthosymmetric sesquilinear form.

First steps going down on algebraic frames

We extend the ring-theoretic concept of going down to algebraic frames and coherent maps. We then use the notion introduced to characterize algebraic frames of dimension 0 and frames of dimension at most 1. An application to rings yields a characterization of von Neumann regular rings that appears to have hitherto been overlooked. Namely, a commutative ring A with identity is von Neumann regular if and only if Ann(I)+P = A, for every prime ideal P of A and any finitely generated ideal I of A contained in P.

Almost slender rings and compact rings

Classical results concerning slenderness for commutative integral domains are generalized to commutative rings with zero divisors. This is done by extending the methods from the domain case and bringing them in connection with results on the linear topologies associated to non-discrete Hausdorff filtrations. In many cases a weakened notion “almost slenderness” of slenderness is appropriate for rings with zero divisors. Special results for countable rings are extended to rings said to be of “bounded type” (including countable rings, ‘small’ rings, and, for instance, rings that are countably generated as algebras over an Artinian ring).More precisely, for a ring R of bounded type it is proved that R is slender if R is reduced and has no simple ideals, or if R is Noetherian and has no simple ideals; moreover, R is almost slender if R is not perfect (in the sense of H. Bass). We use our methods to study various special classes of rings, for instance von Neumann regular rings and valuation rings. Among other results we show that the following two rings are slender: the ring of Puiseux series over a field and the von Neumann regular ring kN/k(N) over a von Neumann regular ring k.For a Noetherian ring R we prove that R is a finite product of local complete rings iff R satisfies one of several (equivalent) conditions of algebraic compactness. A 1-dimensional Noetherian ring is outside this ‘compact’ class precisely when it is almost slender. For the rings of classical algebraic geometry we prove that a localization of an algebra finitely generated over a field is either Artinian or almost slender. Finally, we show that a Noetherian ring R is a finite product of local complete rings with finite residue fields exactly when there exists a map of R-algebras RN→R vanishing on R(N).

On (A)-rings and strong (A)-rings issued from amalgamations

A ring R has the (A)-property (resp., strong (A)-property) if every finitely generated ideal of R consisting entirely of zero divisors (resp., every finitely generated ideal of R generated by a finite number of zero-divisors elements of R) has a nonzero annihilator. The class of commutative rings with property (A) is quite large; for example, Noetherian rings, rings whose prime ideals are maximal, the polynomial ring R[x] and rings whose total ring of quotients are von Neumann regular. Let f : A → B be a ring homomorphism and J be an ideal of B. In this paper, we investigate when the (A)-property and strong (A)-property are satisfied by the amalgamation of rings denoted by A ⋈fJ, introduced by D'Anna, Finocchiaro and Fontana in [3]. Our aim is to construct new original classes of (A)-rings that are not strong (A)-rings, (A)-rings that are not Noetherian and (A)-rings whose total ring of quotients are not Von Neumann regular rings.

Graded Steinberg algebras and their representations

We study the category of left unital graded modules over the Steinberg algebra of a graded ample Hausdorff groupoid. In the first part of the paper, we show that this category is isomorphic to the category of unital left modules over the Steinberg algebra of the skew-product groupoid arising from the grading. To do this, we show that the Steinberg algebra of the skew product is graded isomorphic to a natural generalisation of the the Cohen-Montgomery smash product of the Steinberg algebra of the underlying groupoid with the grading group. In the second part of the paper, we study the minimal (that is, irreducible) representations in the category of graded modules of a Steinberg algebra, and establish a connection between the annihilator ideals of these minimal representations, and effectiveness of the groupoid. Specialising our results, we produce a representation of the monoid of graded finitely generated projective modules over a Leavitt path algebra. We deduce that the lattice of order-ideals in the $K_0$-group of the Leavitt path algebra is isomorphic to the lattice of graded ideals of the algebra. We also investigate the graded monoid for Kumjian--Pask algebras of row-finite $k$-graphs with no sources. We prove that these algebras are graded von Neumann regular rings, and record some structural consequences of this.