Pseudo-rank Function Research Articles

Uniqueness of the von Neumann Continuous Factor

AbstractFor a division ring D, denote by 𝓜D the D-ring obtained as the completion of the direct limit with respect to themetric induced by its unique rank function. We prove that, for any ultramatricial D-ring 𝓑 and any non-discrete extremal pseudo-rank function N on 𝓑, there is an isomorphism of D-rings , where stands for the completion of 𝓑 with respect to the pseudo-metric induced by N. This generalizes a result of von Neumann. We also show a corresponding uniqueness result for *-algebras over fields F with positive definite involution, where the algebra МF is endowed with its natural involution coming from the *-transpose involution on each of the factors .

Isometrical Embeddings of Lattices into Geometric Lattices

A lattice is said to be finite height generated if it is complete and every element is the join of some elements of finite height. Extending former results by Gratzer and Kiss (Order 2:351–365, 1986) on finite lattices, we prove that every finite height generated algebraic lattice that has a pseudorank function is isometrically embeddable into a geometric lattice.

Pseudo-rank functions on certain lattices

Abstract Pseudo-rank functions on bounded lattices are introduced and their properties are studied. It is shown that if the set of all pseudo-rank functions on a Boolean lattice is nonempty then it is a Choquet simplex.

Pseudo-Rank Functions on Rickart *-rings

Abstract. Pseudo-rank functions on Rickart *-rings are introduced and their propertiesare studied. 1. IntroductionA real valued function Don a lattice Lis called a dimension function if therange of D has either an upper bound or a lower bound and for all a;b 2L,D(a_b) + D(a^b) = D(a) + D(b), see; von Neumann [12] p.58. The theory ofdimension functions is studied in various structures. von Neumann [12] introduceddimensionality in continuous geometries by using perspectivity, whereas Iwamura[6] used the concept of a relation called the p-relation .Kaplansky [8], Murray and von Neumann [11] and others have introduced dimen-sionality in rings of operators by using equivalence of projections. Maeda [10] gener-alized the work of von Neumann [12] and Kaplansky [8] for a certain class of lattices.At the same time Loomis [9] gave an abstract setting to the Murray, von Neumanndimension theory by using complete orthocomplemented lattices. Berberian [2] hasdeveloped theory of dimension functions on the lattice of projections of a nite Baer*-ring. Goodearl [4] developed the dimension theory for a certain class of modules.von Neumann [12], p.231 has introduced the concept of a rank-function on a regularring which generalizes the dimension function. Goodearl [3], [5] has introduced anddeveloped the study of pseudo-rank functions on regular rings, which is a general-ization of rank functions.In this paper we introduce and study the concept of a pseudo-rank function on aRickart *-ring R. We obtain some basic properties of pseudo-rank functions andthe set of all pseudo-rank functions on R, on the lines of Goodearl [5] for Rickart*-rings. The unde ned terms are from Berberian [2] and Birkho [1].

Comparability, Stability, and Completions of Ideals#

In this paper, we study the properties of 1-comparability and stable range one condition on ideals of regular rings, and we use these results to investigate normalized pseudo-rank functions on ideals and N*-complete ideals. These will generalize the corresponding results of Goodearl. Finally, we give some sufficient conditions under which ℙ(I) is compact.

Regular skew group rings

AbstractLet G be a group acting on a ring R. We study the problem of determining necessary and sufficient conditions in order that the skew group ring RG be von Neumann regular. Complete characterizations are given in some particular situations, including the case where all idempotents of R are central. For a regular ring R admitting a G-invariant pseudo-rank function N, with G finite, we obtain a necessary condition for RG being regular in terms of the induced action of G on the N-completion of R.

Rank functions on rings derived from group rings

The ring DG derived from the group ring KG is shown to posses exactly one pseudo-rank function in case G is an infinite, but locally and residually finite, group and K is a field. The function is described naturally as the limit of the sequence of normalized ranks of all approximation matrices. This fact is applied to an investigation of the direct finiteness of the maximal quotient ring of DG .

Embeddings of the Maximal Quotient Ring of a von Neumann Regular Ring into its Completion with Respect to a Rank Function

Let $R$ be a commutative von Neumann regular ring. A pseudorank function on $R$ is characterized by a probability measure on the spectrum of $R$. We exhibit two commutative von Neumann regular rings $R$ and $R’$ with the same spectrum $S$, and two rank functions $N$ and $N’$ on $R$ and $R’$, corresponding to the same probability measure on $S$, and such that the maximal right quotient ring ${Q_{\max }}(R)$ embeds in the $N$-completion $\bar R$ of $R$ while ${Q_{\max }}(R’)$ does not embed in $\overline {R’}$.

Embeddings of the maximal quotient ring of a von Neumann regular ring into its completion with respect to a rank function

Let R R be a commutative von Neumann regular ring. A pseudorank function on R R is characterized by a probability measure on the spectrum of R R . We exhibit two commutative von Neumann regular rings R R and R ′ R’ with the same spectrum S S , and two rank functions N N and N ′ N’ on R R and R ′ R’ , corresponding to the same probability measure on S S , and such that the maximal right quotient ring Q max ( R ) {Q_{\max }}(R) embeds in the N N -completion R ¯ \bar R of R R while Q max ( R ′ ) {Q_{\max }}(R’) does not embed in R ′ ¯ \overline {R’} .

Matrix rings over ∗-regular rings and pseudo-rank functions

In this paper we obtain a characterizati on of those * -regular rings whose matrix rings are *-regular satisfying LP ~ RP. This result allows us to obtain a structure theorem for the *-regular self-injective rings of type I which satisfy LP ~ RP matricially. Also, we are concerned with pseudo-rank functions and their corresponding metric completions. We show, amongst other things, that the LP ~ RP axiom extends from a unit-regular * -regular ring to its completion with respect to a pseudo-rank function. Finally, we show that the property LP ~ RP holds for some large classes of *-regular self-injective rings of type II. All rings in this paper are associative with 1. Let R be a ring with an involution *. Recall that * is said to be n-positiυe definite if Σ^x X/X* = 0 implies xλ = = xn = 0. The involution * is said to be proper if it is 1-positive definite; and if * is ^-definite positive for all n, then we say that * is positive definite. Recall than an element e e R is said to be a projection if e2 = e* = e and R is called a Rickart *-ring if for every x e R there exists a projection e in R generating the right annihilator of x, that is t(x) = eR. Because of the involution, we have £(x) = Rf for some projection /. Notice that t(x) Π x*R = 0, hence the involution * is proper and R is nonsingular. The above projections e, f depend on x only, 1 — e (1 — f) is called the right (left) projection of x and, as usual, we shall write 1 - e = RP(JC), 1 -/= LP(JC).

Pseudo-rank functions on skew group rings and on fixed subrings of automorphisms of unit-regular rings

Pseudo-rank functions on skew group rings and on fixed subrings of automorphisms of unit-regular rings

Centres of Rank-Metric Completions

In this paper, we are primarily concerned with the behaviour of the centre with respect to the completion process for von Neumann regular rings at the pseudo-metric topology induced by a pseudo-rank function.Let R be a (von Neumann) regular ring, and N a pseudo-rank function (all terms left undefined here may be found in [6]). Then N induces a pseudo-metric topology on R, and the completion of R at this pseudo-metric, , is a right and left self-injective regular ring. Let Z( ) denote the centre of whatever ring is in the brackets. We are interested in the map .If R is simple, Z(R) is a field, so is discrete in the topology; yet Goodearl has constructed an example with Z(R) = R and Z(R) = C [5, 2.10]. There is thus no hope of a general density result.

Pseudorank functions on crossed products of finite groups over regular rings

Pseudorank functions on crossed products of finite groups over regular rings

TheN *-metric completion of regular rings

We have two main objectives in this paper. Let R be a unit regular ring (the standard reference is [G1]); R admits a pseudo-metric topology, and the completion, denoted S, is called the N*-completion of R. We study the relations between R and S, complementary to a recent article of Goodearl on N*-complete regular rings. The rings R, S are representable as rings of sections of a sheaf-like object, and we clarify and extend this notion. The collection of pseudo-rank functions of R (terms not defined here, will be found in [G1]), denoted IP(R), is a compact convex set, with extreme boundary denoted either OelP(R) or E. We define N* :R--*[0, 1] via N*(r) =sup{P(r)lPslP(R)}. Then N* induces a pseudo-metric topology on R by means of d(r, s)= N*(r-s). This is a metric precisely when N*(r)=0 implies r = 0 ("R is N*-torsion-free"; in [GH2, p. 208], "R has a Hausdorff family of pseudorank functions"); we shall consider only such rings. Chapter 1 deals with properties of the N*-completion of a general (unit regular) ring R, called S. Our main result here (which takes the bulk of the chapter to establish) is that S possesses no new pseudo-rank functions (i.e., the natural map IP(S)~IP(R) is an affine homeomorphism). (This was claimed in [H1, Proposition 15] but the injectivity part of the proof has a gap.) It follows that S is complete in its intrinsic N*-metric and so the recent results of Goodearl I-G2] apply to S. When the appropriate identifications are made, the sup-norm completion of the image of Ko(R) in AfflP(R) [the affine continuous functions on IP(R)] is just Ko(S ). Some consequences of these results include: (i) If K is a metrizable Choquet simplex, there is an N*-complete regular ring S such that Ko(S ) is order-isomorphic to Aft(K). (ii) If(G, u) is an unperforated interpolation group with order unit (see [EHS]), then the sup-norm closure of its image in its natural representation as a group of affine functions on a simplex, is an interpolation group.

Metrically Complete Regular Rings

This paper is concerned with the structure of those (von Neumann) regular rings $R$ which are complete with respect to the weakest metric derived from the pseudo-rank functions on $R$, known as the ${N^ \ast }$-metric. It is proved that this class of regular rings includes all regular rings with bounded index of nilpotence, and all ${\aleph _0}$-continuous regular rings. The major tool of the investigation is the partially ordered Grothendieck group ${K_0}(R)$, which is proved to be an archimedean normcomplete interpolation group. Such a group has a precise representation as affine continuous functions on a Choquet simplex, from earlier work of the author and D. E. Handelman, and additional aspects of its structure are derived here. These results are then translated into ring-theoretic results about the structure of $R$. For instance, it is proved that the simple homomorphic images of $R$ are right and left self-injective rings, and $R$ is a subdirect product of these simple self-injective rings. Also, the isomorphism classes of the finitely generated projective $R$-modules are determined by the isomorphism classes modulo the maximal two-sided ideals of $R$. As another example of the results derived, it is proved that if all simple artinian homomorphic images of $R$ are $n \times n$ matrix rings (for some fixed positive integer $n$), then $R$ is an $n \times n$ matrix ring.

Metrically complete regular rings

This paper is concerned with the structure of those (von Neumann) regular rings $R$ which are complete with respect to the weakest metric derived from the pseudo-rank functions on $R$, known as the ${N^ \ast }$-metric. It is proved that this class of regular rings includes all regular rings with bounded index of nilpotence, and all ${\aleph _0}$-continuous regular rings. The major tool of the investigation is the partially ordered Grothendieck group ${K_0}(R)$, which is proved to be an archimedean normcomplete interpolation group. Such a group has a precise representation as affine continuous functions on a Choquet simplex, from earlier work of the author and D. E. Handelman, and additional aspects of its structure are derived here. These results are then translated into ring-theoretic results about the structure of $R$. For instance, it is proved that the simple homomorphic images of $R$ are right and left self-injective rings, and $R$ is a subdirect product of these simple self-injective rings. Also, the isomorphism classes of the finitely generated projective $R$-modules are determined by the isomorphism classes modulo the maximal two-sided ideals of $R$. As another example of the results derived, it is proved that if all simple artinian homomorphic images of $R$ are $n \times n$ matrix rings (for some fixed positive integer $n$), then $R$ is an $n \times n$ matrix ring.

Representing Rank Complete Continuous Rings

Given a suitable regular ring R, we construct a sheaf-like representation for R as a ring of continuous sections from a completely regular space to an appropriately toplogized disjoint union of factor rings corresponding to ‘'extremal“ pseudo-rank functions. Applied to rings which are complete with respect to a rank function this representation is an isomorphism, the completely regular space is extremally disconnected and compact, and the * ‘stalks” are the simple factor rings.

Rank functions and K 0 of regular rings

This paper is concerned with the existence and uniqueness of rank and pseudo-rank functions on a von Neumann regular ring R. The main technique used involves transferring hypotheses becomes a partially ordered abelian group. It is shown that the existence of a pseudo-rank function on R is equivalent to certain finiteness conditions on the matrix rings over R. As a corollary, necessary and sufficient conditions are obtained for the existence of a rank function on a simple regular ring. Uniqueness of a rank function is shown to be equivalent to certain comparability conditions on the principal right ideals of R. Other results concern the existence of enough pseudo-rank functions to distinguish nonzero ring elements from zero, or to distinguish between non-isomorphic principal right ideals. All rings in this paper are associative with unit (but usually noncommutative), and all modules are unital right modules. We use “regular” to mean “von Neumann regular”. The research of the first author was partially supported by National Science Foundation Grant No. GP-43029.