On Strongly Clean Matrix and Triangular Matrix Rings

It is known that structural matrix rings pro-vide a natural passage from complete matrix rings to upper and lower triangular matrix rings, and they often explain the peculiarities regarding certain properties of complete matrix rings on the one hand and of triangular matrix rings on the other hand. In this paper the concept of a set of matrix units in a ring associated with a quasi-order relation is introduced and used to provide an internal char-acterisation of structural matrix rings.

Let R be an associative ring with identity 1. We describe all idempotent matrices with only zeros and ones on the diagonal in – the ring of upper triangular matrices over R (), and – the ring of infinite upper triangular matrices (indexed by ) over R. Moreover, when R is finite, we calculate the number of all idempotent matrices with only zeros and ones on the diagonal in .

In this paper, we introduce the concept of a triangular coefficient matrix ring and investigate the structure of its ideals.We then characterize the radicals of the ring R h [x]/x n for every positive integer n, where R h [x] denotes the Hurwitz polynomial ring and x n represents the ideal of this ring generated by x n .Furthermore, we explore several properties that are transferred between the base ring R and the matrix ring Hn(R), which is a proper subring of the triangular coefficient matrix ring.

In 1963, Gilmer characterized all finite commutative rings with a cyclic group of units and, in 1967, Eldridge and Fischer generalized these results to rings with minimum condition. In the present paper these results are extended to semiperf ect rings and generalizations of the three theorems are obtained. It is shown that a semiperfect ring with cyclic group of units is finite and is either commutative or is the direct sum of a commutative ring and the 2 x 2 upper triangular matrix ring over the field of two elements. Let R be semiperfect with an abelian group of units. It is shown that R is finite if either the group of units is finite or the group of units is finitely generated and the Jacobson radical is nil. The proofs of all these results depend on our main theorem: The structure of a semiperfect ring R with an abelian group of units is described completely up to the structure of commutative local rings. (That is commutative rings with a unique maximal ideal.) The groups of units of these local rings are shown to be direct factors of the group of units of R.

The article investigates the classification with precision up to equivalence of involutions in the algebra of upper triangular matrices over the ring of integers of algebraic numbers of quadratic fields.The description of involutions in algebras represents one of the classical problems of ring theory. Standard examples of involutions are transposition in matrix algebra and conjugation in the field of complex numbers and the algebra of quaternions.In the case where the field 𝑃 has a characteristic different from two, a complete description of involutions with precision up to their equivalence in the algebra 𝑇𝑛(𝑃) for any natural number 𝑛 was obtained in [15]. In this work [3] involutions in the algebra of upper triangular matrices over commutative rings are studied. If the ring is a field of characteristic 2 or a Boolean ring, then necessary and sufficient conditions for the finiteness of the number of equivalence classesof involutions were found.This article is a continuation of the work of [3]. In the article [3], in particular, the number of equivalence classes of involutions in the algebras of upper triangular matrices over the ring of integers was found. In this regard, the natural result is the problem of describing involutions with precision up to their equivalence in algebras of upper triangular matrices over the ring of algebraic integers of quadratic fields, to which this work is devoted. In the work, the number of equivalence classes of involutions in such algebras is found and the method of finding representatives in each equivalence class is illustrated with examples. Upon receipt the main results in this work, the apparatus of the theory of Pell’s equations is significantly used.

In this article, we study several probabilistic properties of polynomials defined over the ring of p-adic integers under the Haar measure. First, we calculate the probability that a monic polynomial is separable, generalizing a result of Polak. Second, we introduce the notion of two polynomials being strongly coprime and calculate the probability of two monic polynomials being strongly coprime. Finally, we explain how our method can be used to extrapolate other probabilistic properties of polynomials over the ring of p-adic integers from polynomials defined over the integers modulo powers of p.

Strong cleanness of the [formula omitted] matrix ring over a general local ring

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.

Standard techniques are used to compute K i ( i = 0 , 1 , 2 ) {K_i}(i = 0,1,2) of generalized triangular matrix rings.

In this paper we study the R -braces $$(M,+,\circ )$$ ( M , + , ∘ ) such that $$M\cdot M$$ M · M is cyclic, where R is the ring of p -adic integers and $$\cdot $$ · is the product of the commutative and 3-nilpotent R -algebra associated to M . In particular, we give a classification up to isomorphism in the torsion-free case and up to isoclinism in the torsion case. More precisely, the isomorphism classes and the isoclinism classes of such algebras are in correspondence with particular equivalence classes of the bilinear forms defined starting from the products of the algebras.

We show how basic orders for defect two blocks of symmetric groups over the ring of p-adic integers can be constructed by purely combinatorial means.

Theory of dynamical systems in fields of p-adic numbers is an important part of algebraic and arithmetic dynamics. The study of p-adic dynamical systems is motivated by their applications in various areas of mathematics, e.g., in physics, genetics, biology, cognitive science, neurophysiology, computer science, cryptology, etc.In particular, p-adic dynamical systems found applications in cryptography, which stimulated the interest to nonsmooth dynamical maps. An important class of (in general) nonsmooth maps is given by 1-Lipschitz functions.In this thesis we restrict our study to the class of 1-Lipschitz functions and describe measure-preserving (for the Haar measure on the ring of p-adic integers) and ergodic functions.The main mathematical tool used in this work is the representation of the function by the van der Put series which is actively used in p-adic analysis. The van der Put basis differs fundamentally from previously used ones (for example, the monomial and Mahler basis) which are related to the algebraic structure of p-adic fields. The basic point in the construction of van der Put basis is the continuity of the characteristic function of a p-adic ball.Also we use an algebraic structure (permutations) induced by coordinate functions with partially frozen variables.In this thesis, we present a description of 1-Lipschitz measure-preserving and ergodic functions for arbitrary prime p.

Supergeometry and arithmetic geometry

Riesz products on the ring of p-adic integers are introduced and studied from the points of view of harmonic analysis and dynamical system relative to the shift transformation. We find necessary and sufficient conditions for a Riesz product to be invariant, quasi-invariant or quasi-Bernoulli. We study the mutual absolute continuity of two Riesz products and the almost everywhere convergence of lacunary series with respect to a Riesz product. We also compute the Hausdorff dimension and the energy dimension and prove a multifractal formalism for a given Riesz product.

Relation between the functors Hom and Tor over the ring of p-adic integers