Von Neumann Regular Research Articles

On the Geometry of the Unit Ball of aJB*-Triple

We explore aJB*-triple analogue of the notion of quasi invertible elements, originally studied by Brown and Pedersen in the setting ofC*-algebras. This class of BP-quasi invertible elements properly includes all invertible elements and all extreme points of the unit ball and is properly included in von Neumann regular elements in aJB*-triple; this indicates their structural richness. We initiate a study of the unit ball of aJB*-triple investigating some structural properties of the BP-quasi invertible elements; here and in sequent papers, we show that various results on unitary convex decompositions and regular approximations can be extended to the setting of BP-quasi invertible elements. SomeC*-algebra andJB*-algebra results, due to Kadison and Pedersen, Rørdam, Brown, Wright and Youngson, and Siddiqui, including the Russo-Dye theorem, are extended toJB*-triples.

ON GI-FLAT MODULES AND DIMENSIONS

Let R be a ring. A right R-module M is called GI-flat if <TEX>$Tor^R_1(M,G)=0$</TEX> for every Gorenstein injective left R-module G. It is shown that GI-flat modules lie strictly between flat modules and copure flat modules. Suppose R is an <TEX>$n$</TEX>-FC ring, we prove that a finitely presented right R-module M is GI-flat if and only if M is a cokernel of a Gorenstein flat preenvelope K <TEX>${\rightarrow}$</TEX> F of a right R-module K with F flat. Then we study GI-flat dimensions of modules and rings. Various results in [6] are developed, some new characterizations of von Neumann regular rings are given.

MONOIDS OVER WHICH ALL REGULAR RIGHT S-ACTS ARE WEAKLY INJECTIVE

There have been some study characterizing monoids by homological classification using the properties around projectivity, injectivity, or regularity of acts. In particular Kilp and Knauer([4]) have analyzed monoids over which all acts with one of the properties around projectivity or injectivity are regular. However Kilp and Knauer left over problems of characterization of monoids over which all regular right S-acts are (weakly) at, (weakly) injective or faithful. Among these open problems, Liu([3]) proved that all regular right S-acts are (weakly) at if and only if es is a von Neumann regular element of S for all <TEX>$s{\in}S$</TEX> and <TEX>$e^2=e{\in}T$</TEX>, and that all regular right S-acts are faithful if and only if all right ideals eS, <TEX>$e^2=e{\in}T$</TEX>, are faithful. But it still remains an open question to characterize over which all regular right S-acts are weakly injective or injective. Hence the purpose of this study is to investigate the relations between regular right S-acts and weakly injective right S-acts, and then characterize the monoid over which all regular right S-acts are weakly injective.

SOME RESULTS ON TORSIONFREE MODULES

We study torsionfree and divisible dimensions in terms of right derived functors of -⊗-. We also investigate the cotorsion pair cogenerated by the class of cyclic torsionfree right R-modules. As applications, some new characterizations of von Neumann regular rings, F-rings and semisimple Artinian rings are given.

On Modules Over Group Rings

Let M be a right module over a ring R and let G be a group. The set MG of all formal finite sums of the form ∑ g ∈ Gmgg where mg ∈ M becomes a right module over the group ring RG under addition and scalar multiplication similar to the addition and multiplication of a group ring. In this paper, we study basic properties of the RG-module MG, and characterize module properties of (MG)RG in terms of properties of MR and G. Particularly, we prove the module-theoretic versions of several well-known results on group rings, including Maschke’s Theorem and the classical characterizations of right self-injective group rings and of von Neumann regular group rings.

Von Neumann Regular and Related Elements in Commutative Rings

Let R be a commutative ring with nonzero identity. In this paper, we study the von Neumann regular elements of R. We also study the idempotent elements, π-regular elements, the von Neumann local elements, and the clean elements of R. Finally, we investigate the subgraphs of the zero-divisor graph Γ(R) of R induced by the above elements.

A note on complete rings of quotients and McCoy rings

If a (commutative unital) ring \(A\) is reduced and coincides with its total quotient ring, then \(A\) satisfies Property A (that is, \(A\) is a McCoy ring) if and only if the inclusion of \(A\) in its complete ring of quotients \(C(A)\) is a survival extension. The “if” assertion fails if one deletes the hypothesis that \(A\) is reduced. This is shown by using the idealization construction to construct a suitable ring \(A\) and then identifying its complete ring of quotients (which turns out to be a related idealization). Related characterizations of von Neumann regular rings are also given with the aid of the going-down property GD of ring extensions. For instance, a ring \(A\) is von Neumann regular if and only if \(A\) is a reduced McCoy ring that coincides with its total quotient ring such that \(A \subseteq C(A)\) satisfies GD.

NIL-ARMENDARIZ RINGS AND UPPER NILRADICALS

We continue the study of nil-Armendariz rings, initiated by Antoine, and Armendariz rings. We first examine a kind of ring coproduct constructed by Antoine for which the Armendariz, nil-Armendariz, and weak Armendariz properties are equivalent. Such a ring has an important role in the study of Armendariz ring property and near-related ring properties. We next prove an Antoine's result in relation with the ring coproduct by means of a simpler direct method. In the proof we can observe the concrete shapes of coefficients of zero-dividing polynomials. We next observe the structure of nil-Armendariz rings via the upper nilradicals. It is also shown that a ring R is Armendariz if and only if R is nil-Armendariz if and only if R is weak Armendariz, when R is a von Neumann regular ring.

Ascending the divided and going-down properties by absolute flatness

This paper aims to show that the “going-down ring” and the “divided ring” properties ascend along flat morphisms whose co-diagonal morphisms are flat, the so-called absolutely flat morphisms introduced by Olivier. But unibranchedness hypotheses are necessary as any henselization morphism shows. As a by-product, we get that the “unibranched divided ring” property is preserved by the formation of factor domains and by localization with respect to prime ideals. Moreover, we exhibit some ascent results for the “going-down ring” and “divided ring” properties along integral extensions. Open image in new window

Takeuchi’s free Hopf algebra construction revisited

Takeuchi’s famous free Hopf algebra construction is analyzed from a categorical point of view, and so is the construction of the Hopf envelope of a bialgebra. Both constructions in fact can be described as compositions of well known and natural constructions. This way certain partially wrong perceptions of these constructions are clarified and their mutual relation is made precise. The construction of Hopf envelopes finally is shown to provide a construction of a Hopf coreflection of bialgebras by simple dualization. The results provided hold for any commutative von Neumann regular ring, not only for fields.

On p.p. structural matrix rings

A ring is called a left p.p. ring if every principal left ideal is projective. The objective here is to completely determine the left p.p. structural matrix rings over a von Neumann regular ring.

Direct sums of Rickart modules

The notion of Rickart modules was defined recently. It has been shown that a direct sum of Rickart modules is not a Rickart module, in general. In this paper we investigate the question: When are the direct sums of Rickart modules, also Rickart? We show that if M i is M j -injective for all i < j ∈ I = { 1 , 2 , … , n } then ⊕ i = 1 n M i is a Rickart module if and only if M i is M j -Rickart for all i , j ∈ I . As a consequence we obtain that for a nonsingular extending module M, E ( M ) ⊕ M is always a Rickart module. Other characterizations for direct sums to be Rickart under certain assumptions are provided. We also investigate when certain classes of free modules over a ring R, are Rickart. It is shown that every finitely generated free R-module is Rickart precisely when R is a right semihereditary ring. As an application, we show that a commutative domain R is Prüfer if and only if the free R-module R ( 2 ) is Rickart. We exhibit an example of a module M for which M ( 2 ) is Rickart but M ( 3 ) is not so. Further, von Neumann regular rings are characterized in terms of Rickart modules. It is shown that the class of rings R for which every finitely cogenerated right R-module is Rickart, is precisely that of right V-rings. Examples which delineate the concepts and the results are provided.

On the torsion graph and von Neumann regular rings

Let R be a commutative ring with identity and M be a unitary R-module. A torsion graph of M, denoted by ?(M), is a graph whose vertices are the non-zero torsion elements of M, and two distinct vertices x and y are adjacent if and only if [x : M][y : M]M = 0. In this paper, we investigate the relationship between the diameters of ?(M) and ?(R), and give some properties of minimal prime submodules of a multiplication R-module M over a von Neumann regular ring. In particular, we show that for a multiplication R-module M over a B?zout ring R the diameter of ?(M) and ?(R) is equal, where M , T(M). Also, we prove that, for a faithful multiplication R-module M with |M|?4,?(M) is a complete graph if and only if ?(R) is a complete graph.

Commutative Boolean monoids, reduced rings, and the compressed zero-divisor graph

Let R be a commutative ring with 1≠0. The zero-divisor graph Γ(R) of R is the (undirected) graph whose vertices consist of the nonzero zero-divisors of R such that distinct vertices x and y are adjacent if and only if xy=0. The relation on R given by r∼s if and only if annR(r)=annR(s) is an equivalence relation. The compressed zero-divisor graph ΓE(R) is the (undirected) graph whose vertices are the equivalence classes induced by ∼ other than [0] and [1], such that distinct vertices [r] and [s] are adjacent in ΓE(R) if and only if rs=0. We investigate ΓE(R) when R is reduced and are interested in when ΓE(R)≅Γ(S) for a reduced ring S. Among other results, it is shown that ΓE(R)≅Γ(B) for some Boolean ring B if and only if Γ(R) (and hence ΓE(R)) is a complemented graph, and this is equivalent to the total quotient ring of R being a von Neumann regular ring.

A GENERALIZATION OF PRÜFER'S ASCENT RESULT TO NORMAL PAIRS OF COMPLEMENTED RINGS

Let R ⊆ T be a (unital) extension of (commutative) rings, such that the total quotient ring of R is a von Neumann regular ring and T is torsion-free as an R-module. Let T ⊆ B be a ring extension such that B is a reduced ring that is torsion-free as a T-module. Let R* (respectively, A) be the integral closure of R in T (respectively, in B). Then (R*, T) is a normal pair (i.e. S is integrally closed in T for each ring S such that R* ⊆ S ⊆ T) if and only if (A, AT) is a normal pair. This generalizes results of Prüfer and Heinzer on Prüfer domains to normal pairs of complemented rings.

Dual Rickart Modules

Rickart property for modules has been studied recently. In this article, we introduce and study the notion of dual Rickart modules. A number of characterizations of dual Rickart modules are provided. It is shown that the class of rings R for which every right R-module is dual Rickart is precisely that of semisimple artinian rings, the class of rings R for which every finitely generated free R-module is dual Rickart is exactly that of von Neumann regular rings, while the class of rings R for which every injective R-module is dual Rickart is precisely that of right hereditary ones. We show that the endomorphism ring of a dual Rickart module is always left Rickart and obtain conditions for the converse to hold true. We prove that a dual Rickart module with no infinite set of nonzero orthogonal idempotents in its endomorphism ring is a dual Baer module. A structure theorem for a finitely generated dual Rickart module over a commutative noetherian ring is provided. It is shown that, while a direct summand of a dual Rickart module inherits the property, direct sums of dual Rickart modules do not. We introduce the notion of relative dual Rickart property and show that if M i is M j -projective for all i > j ∈ ℐ = {1, 2,…, n} then is a dual Rickart module if and only if M i is M j -d-Rickart for all i, j ∈ ℐ. Other instances of when a direct sum of dual Rickart modules is dual Rickart, are included. Examples which delineate the concepts and results are provided.

Prime Submodules of Regular Modules Over Commutative Rings

We investigate prime submodules of regular modules. In particular, we shall see that every proper submodule of a regular module M over an integral domain is prime. Although (von Neumann) regular rings are zero dimensional (primes ideals are maximal), it is not true for regular modules in general. We show that cyclic regular module are zero dimensional. Furthermore, it is proved that for a regular module every proper submodule is a radical submodule.

On Subcategories of the Category of Hopf Algebras

The various canonical subcategories of the category HopfR of Hopf algebras over a commutative ring R, like those of (co)commutative Hopf algebras or Hopf algebras whose antipode is bijective or of order 2, are shown to be locally presentable categories and reflective and coreflective in their respective supercategories. The reflectivity results provided only hold for commutative von Neumann regular rings, while most of the coreflectivity results are valid over any ring. As a consequence one gets existence of free commutative Hopf algebras over coalgebras and cofree cocommutative Hopf algebras over algebras.

RINGS OVER WHICH COEFFICIENTS OF NILPOTENT POLYNOMIALS ARE NILPOTENT

Antoine studied conditions which are connected to the question of Amitsur of whether or not a polynomial ring over a nil ring is nil, observing the structure of nilpotent elements in Armendariz rings and introducing the notion of nil-Armendariz rings. The class of nil-Armendariz rings contains Armendariz rings and NI rings. We continue the study of nil-Armendariz rings, concentrating on the structure of rings over which coefficients of nilpotent polynomials are nilpotent. In the procedure we introduce the notion of CN-rings that is a generalization of nil-Armendariz rings. We first construct a CN-ring but not nil-Armendariz. This may be a base on which Antoine's theory can be applied and elaborated. We investigate basic ring theoretic properties of CN-rings, and observe various kinds of CN-rings including ordinary ring extensions. It is shown that a ring R is CN if and only if R is nil-Armendariz if and only if R is Armendariz if and only if R is reduced when R is a von Neumann regular ring.

Feedback classification of linear systems over von Neumann regular rings

It is proved that feedback classification of a linear system over a commutative von Neumann regular ring R can be reduced to the classification of a finite family of systems, each of which is properly split into a reachable and a non-reachable part, where the reachable part is in a Brunovski-type canonical form, while the non-reachable part can only be altered by similarity. If a canonical form is known for similarity of matrices over R, then it can be used to construct a canonical form for feedback equivalence. An explicit algorithm is given to obtain the canonical form in a computable context together with an example over a finite ring.