On von Neumann regular rings, III

A commutative Noetherian local ring of finite global dimension is a regular local ring and the structure of such a ring is well described 1n many texts (see §1.9). It is perhaps surprising that very little seems to be known about the corresponding non-commutat1ve rings and it is our aim 1n this thesis to examine some of the ways in which the commutative theory can, and cannot be extended to a right Noetherian local ring of finite right global dimension. Chapter 1 contains basic definitions and, in Chapter 2, results are obtained on the projective and Injective dimensions of modules over right Noetherian local rings. Commutative regular local rings are domains and we begin Chapter 3 by considering the question of when a right Noetherian local ring R of finite right global dimension is a prime ring. An example of Stafford's shows that R is not necessarily prime; however, by examining the Nilpotent radical of R, we are able to extend results of Ramras [57] and Walker [75] and show that R is indeed prime when certain prime ideals are localisable. In Chapter 4 we consider the lattice of prime ideals of R when R is an AR-r1ng and provide theorems which generalise some of the basic results from the theory of commutative regular local rings. Section 4.3 contains examples which not only illustrate points arising in Chapters 3 and 4 but also show that some of the techniques which are mainstays of the commutative theory, fall dramatically in a non-commutat1ve setting. In Chapter 5, we generalise the concepts of regular sequences and Cohen Macaulay rings enabling us to prove, in §5.2 and chapter 6, that a right Noetherian local ring of finite right global dimension, which 1s integral over its centre, is a prime ring and exhibits many properties similar to those enjoyed by commutative regular local rings. Examples are provided which show that some alternative approaches are not applicable to the rings considered in these chapters. Regular local rings are Gorensteln rings and, as such, are the subject of an elegant structure theorem due to Bass [5]. In Chapter 7, we generalise his theorem to the situation of rings Integral over their centres. Each chapter begins with a summary of the results contained in that chapter.

Let R be a prime ring with maximal right ring of quotients Q, extended centroid C and a noncommutative Lie ideal L. Our aim is to characterize the forms of generalized skew derivations $$\delta $$ and $$\varrho $$ of R in case of $$\begin{aligned} p(\delta (u)^{l_{1}}\varrho (u)^{l_{2}}\delta (u)^{l_{3}}\varrho (u)^{l_{4}}\cdots \varrho (u)^{l_{k}} )^{n}=0, \end{aligned}$$ for all $$u\in L$$ , where $$0 \ne p\in R$$ , $$l_{1},l_{2},\ldots ,l_{k}$$ are fixed nonnegative integers with $$l_{1}\ne 0$$ and n is a positive fixed integer. The proof of this main result is based on the so-called extended Jacobson density theorem, which is different from that of previous results in the literature. Also as a result, we obtain if $$p\delta (u)^{n}=0$$ for all $$u \in L$$ , then there exists $$a\in Q$$ such that $$\delta (x)=ax$$ for all $$x \in R$$ and $$pa=0$$ unless R satisfies $$s_4$$ .

Following O'Meara's result [Journal of Algebra and Its Applications, 13 (2014) 8], it follows that the block matrix A = B 0 0 0 M n+r (R), B M n (R), r 1, over a von Neumann regular separative ring R, is a product of idempotent matrices.Furthermore, this decomposition into idempotents of A also holds when B is an invertible matrix and R is a GE ring (defined by Cohn [New mathematical monographs: 3, Cambridge University Press, 2006]).As a consequence, it follows that if there exists an example of a von Neumann regular ring R over which the matrix A = B 0 0 0 M n+r (R), where B M n (R), r 1, cannot be expressed as a product of idempotents, then R is not separative, thus providing an answer to an open question whether there exists a von Neumann regular ring which is not separative.The paper concludes with a counter example related to an open question whether every totally nonnegative matrix is a product of nonnegative idempotent matrices.

We give a negative answer to a question posed by A. V. Jategaonkar: is it not true that an arbitrary primary principal left ideal ring is a factor of a prime principal left ideal ring? We give a counter example in the class of finite complete primary principal ideal rings, the so-called Galois-Eisenstein-Ore rings.

Given arbitrary rings A, B such that A ⊂ B and an arbitrary prime ideal P of A, we show how to construct a quotient ring Ap such that A ⊂ Ap ⊂ B. The ring Ap contains a prime ideal P lying over P and the prime ring A/P may be embedded in Ap/P.

Let R be a prime ring and δ a derivation of R. Divided powers \( D_n ^{\underline{\underline {def.}} } \tfrac{1} {{n!}}\tfrac{{d^n }} {{dx^n }} \) of ordinary differentiation d/dx form Hasse-Schmidt higher derivations of the Ore extension (skew polynomial ring) R[x; δ]. They have been used crucially but implicitly in the investigation of R[x; δ]. Our aim is to explore this notion. The following is proved among others: Let Q be the left Martindale quotient ring of R. It is shown that \( S^{\underline{\underline {def.}} } Q[x;\delta ] \) is a quasi-injective (R, R)-module and that any (R,R)-bimodule endomorphism of S can be uniquely expressed in the form $$ \theta (f) = \sum\limits_{n = 0}^\infty {\zeta _n D_n (f)} forf \in Q[x;\delta ], $$ where ζ n ∈ C S (R), the centralizer of R in S. As an application, we also use the Ore extension R[x; δ] to deduce Kharchenko’s theorem for a single derivation. These results are extended to the Ore extension R[X;D] of R by a sequence D of derivations of R.

Let R be a ring. An element p of R is a prime element if pR = Rp is a prime ideal of R. A prime ring R is said to be a Unique Factorisation Ring if every non-zero prime ideal contains a prime element. This paper develops the basic theory of U.F.R.s. We show that every polynomial extension in central indeterminates of a U.F.R. is a U.F.R. We consider in more detail the case when a U.F.R. is either Noetherian or satisfies a polynomial identity. In particular we show that such a ring R is a maximal order, that every height-1 prime ideal of R has a classical localisation in which every two-sided ideal is principal, and that R is the intersection of a left and right Noetherian ring and a simple ring.

The local factorization theorem of Zariski and Abhyankar characterizes all $2$-dimensional regular local rings which lie between a given $2$-dimensional regular local ring $R$ and its quotient field as finite quadratic transforms of $R$. This paper shows that every regular local ring $R$ of dimension $n > 2$ has infinitely many minimal regular local overrings which cannot be obtained by a monoidal transform of $R$. These overrings are localizations of rings generated over $R$ by certain quotients of elements of an $R$-sequence. Necessary and sufficient conditions are given for this type of extension of $R$ to be regular.

This note contains certain conditions for unit-regularity and for selfinjective regular rings to have bounded index (such rings have many interesting properties [9]). Sufficient conditions are also given for rings to be Noetherian, Artinian and quasi-Frobeniusian.

In part I we introduce vertex rings , which bear the same relation to vertex algebras (or VOAs) as commutative, associative rings do to commutative, associative algebras over C \mathbf {C} . We show that vertex rings are characterized by Goddard axioms . These include a generalization of the translation-covariance axiom of VOA theory that involves a canonical Hasse-Schmidt derivation naturally associated to any vertex ring. We give several illustrative applications of these axioms, including the construction of vertex rings associated with the Virasoro algebra. We consider some categories of vertex rings, and the rôle played by the center of a vertex ring. In part II we extend the theory of Pierce bundles associated to a commutative ring to the setting of vertex rings. This amounts to the construction of certain reduced étale bundles of vertex rings functorially associated to a vertex ring. We introduce von Neumann regular vertex rings as a generalization of von Neumann regular commutative rings; we obtain a characterization of this class of vertex rings as those whose Pierce bundles are bundles of simple vertex rings.

An element x of a semigroup [ring] A is said to regular if there is an element y of A such that x = xyx, and A is called a regular semigroup [(Von Neumann) regular ring] if every element of A is regular. A quasi-ideal of a semigroup [ring] A is a subsemigroup [subring] Q of A such that AQ intersection QA [is a subset of] Q,and a bi-ideal of A is a subsemingroup [subring] B of A such that BAB [is a subset of] B. Recall that for nonempty subsets X and Y of a ring A, XY denotes the set of all finite sums of the form sigma x[subscript i] y[subscript i] where x[subscript i] [is an element of set] X and Y[subscript i] [is an element of set] Y. We say that a semigroup or a ring has the BQ-property if its quasi-ideals and bi-ideals coincide. It is know that every regular semigroup and every regular ring has the BQ-property. For a nonempty set X, let T(X) denote the full transformation semigroup on X, and for [is an empty set] is not equal to Y[is a subset of] X, let T (X,Y) and T [bar] (X, Y) be the subsemigroups of T(X) defined by T(X, Y)={ alpha is an element of a set T(X) l ran alpha is a subset of Y} and T[bar] (X, Y)={ alpha is an elemaent of a set T(X) l Yalpha is a subset of Y}. Symons and Magill introduced and studied T(X, Y) and T[bar] (X, Y) in 1975 and 1966, respectively. If V is a vector space over a field F, let L[subscript F](V) be the set of all linear transformations alpha : V vector V. For a subspace W of V, define L[subscript F] (V, W) and L[bar][subscript F] (V, W) analogously as follows : L[subscript F] (V, W) = {alpha is an element of a set L[subscriptF(V) l ran alpha is a subset of W} and L[bar][subscript F](V, W) = {alpha is an element of a set L[subscritp F] (V) l { Walpha is a subset of W},and we also consider K[subscript F](V, W) = {alpha L[subscriptF] (V) l W is a subset of ker alpha}. Then L[subscriptF](V, W), L[bar] [subscript F](V, W) and K[subscript F] (V, W) are subsemigroups of (L[subscript F] (V), omicron) and subrings of (L[subscript F] (V),+,omicron) where omicron and+are the composition and usual addition of linear transformations, respectively. This research consists of two major parts. In the first part, we give necessary and sufficient conditions for the elements of these semigroups to be regular. As a consequence, characterizations determinging when these semigroups are regular are given. In the second part, we provide necessary and sufficient conditions for these semigroups and rings to have the BQ-property.

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be any directed graph, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> any field. We classify those graphs <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which the Leavitt path algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript upper K Baseline left-parenthesis upper E right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>K</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>E</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L_K(E)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is primitive. As a consequence, we obtain classes of examples of von Neumann regular prime rings which are not primitive.

A ring R is called a left (right) SF-ring if all simple left (right) R-modules are flat. It is known that von Neumann regular rings are left and right SF-rings. In this article, we study the regularity of left SF-rings and we prove the following: 1) if R is a left SF-ring whose all complement left (right) ideals are W-ideals, then R is strongly regular; 2) if R is a left SF-ring whose all maximal essential right ideals are GW-ideals, then R is regular.

In this paper, we first introduce and study the notions of [Formula: see text]-flat modules. Then study the rings over which all modules are [Formula: see text]-flat (which is called [Formula: see text]-VN regular rings). In particular, we show that a ring [Formula: see text] is a [Formula: see text]-VN regular ring if and only if T[Formula: see text] is a von Neumann regular ring, if and only if [Formula: see text] is a reduced ring with Min[Formula: see text] compact. Finally, we obtain the Chase theorem for [Formula: see text]-coheret rings: a ring [Formula: see text] is [Formula: see text]-coherent if and only if any direct product of [Formula: see text] is [Formula: see text]-flat if and only if any direct product of flat [Formula: see text]-modules is [Formula: see text]-flat. Some examples are provided to compare with the known conceptions.

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