Stable Range Condition Research Articles

Stable range conditions for abelian and duo rings

The article deals with the following question: when does the classical ring of quotientsof a duo ring exist and idempotents in the classical ring of quotients $Q_{Cl} (R)$ are thereidempotents in $R$? In the article we introduce the concepts of a ring of (von Neumann) regularrange 1, a ring of semihereditary range 1, a ring of regular range 1. We find relationshipsbetween the introduced classes of rings and known ones for abelian and duo rings.We proved that semihereditary local duo ring is a ring of semihereditary range 1. Also it was proved that a regular local Bezout duo ring is a ring of stable range 2. In particular, the following Theorem 1 is proved: For an abelian ring $R$ the following conditions are equivalent:$1.$\ $R$ is a ring of stable range 1; $2.$\ $R$ is a ring of von Neumann regular range 1.
 The paper also introduces the concept of the Gelfand element and a ring of the Gelfand range 1 for the case of a duo ring. Weproved that the Hermite duo ring of the Gelfand range 1 is an elementary divisor ring (Theorem 3).

Reverse decomposition of unipotents over noncommutative rings I: General linear groups

Recently it has been proved that if σ∈GLn(R) where R is a commutative ring and n≥3, then each of the matrices tkl(σij)(i≠j,k≠l) is a product of eight En(R)-conjugates of σ and σ−1. In this article we show that similar results hold true if R is a (noncommutative) von Neumann regular ring, or a Banach algebra, or a ring satisfying a stable range condition, or a ring with Euclidean algorithm, or an almost commutative ring.

Injective stability for odd-dimensional unitary K1{K_{1}}

Abstract In this paper, under the usual stable range condition, a decomposition theorem for the elementary subgroup is obtained, and the injective stability theorem for odd-dimensional unitary K 1 {K_{1}} is proved.

Stable range one on non-zero divisors

In this paper, almost unit 1-stable range condition for a commutative ring with unity is defined and similar results which are already proved for the weakly unit 1-stable range are proved. Several properties of rings satisfying almost unit 1-stable range are discussed. The characterizations for rings such as presimplifiable semilocal rings and idealization R(M) of a ring R and an R-module M, satisfying almost unit 1-stable range is also given in this paper. Few examples are given to show that almost unit 1-stable range condition is different from other stable range conditions.

A NOTE ON STRONGLY *-CLEAN RINGS

A *-ring R is called (strongly) *-clean if every element of R is the sum of a projection and a unit (which commute with each other). In this note, some properties of *-clean rings are considered. In particular, a new class of *-clean rings which called strongly <TEX>${\pi}$</TEX>-*-regular are introduced. It is shown that R is strongly <TEX>${\pi}$</TEX>-*-regular if and only if R is <TEX>${\pi}$</TEX>-regular and every idempotent of R is a projection if and only if R/J(R) is strongly regular with J(R) nil, and every idempotent of R/J(R) is lifted to a central projection of R. In addition, the stable range conditions of *-clean rings are discussed, and equivalent conditions among *-rings related to *-cleanness are obtained.

Normality and [formula omitted]-stability of Roy's elementary orthogonal group

In this paper, we prove the normality of the elementary orthogonal group (Dickson–Siegel–Eichler–Roy or DSER group) over a commutative ring which was introduced by A. Roy in [14] under some conditions on the hyperbolic rank. We also establish stability theorems for K1 of Roy's group under different stable range conditions. We obtain a decomposition theorem for the elementary orthogonal group which is used to deduce the stability theorem.

Injective stability for unitary K1, revisited

AbstractWe prove the injective stability theorem for unitary K1 under the usual stable range condition on the ground ring. This improves the stability theorem of A. Bak, V. Petrov and G. Tang where a stronger Λ-stable range condition was used.

Commutative domains of elementary divisors and some properties of their elements

We study commutative domains of elementary divisors from the viewpoint of investigation of the structure of invertible matrices that reduce a given matrix to the diagonal form. Some properties of elements of these domains are indicated. We establish conditions, close to the stable-range conditions, under which a commutative Bezout domain is a domain of elementary divisors.

The Monogenic Fischer Decomposition: Two Vector Variables

In this paper we will present two proofs of the monogenic Fischer decomposition in two vector variables. The first one is based on the so-called “Harmonic Separation of Variables Theorem” while the second one relies on some simple dimension arguments. We also show that these decomposition are still valid under milder assumptions than the usual stable range condition. In the process, we derive explicit formula for the summands in the monogenic Fischer decomposition of harmonics.

Clean matrices over commutative rings

A matrix $A\in M_n(R)$ is $e$-clean provided there exists an idempotent $E\in M_n(R)$ such that $A-E\in \mathop{\rm GL}_n(R)$ and $\det E=e$. We get a general criterion of $e$-cleanness for the matrix $[[a_1,a_2,\cdots ,a_{n+1}]]$. Under the $n$-stable range condition, it is shown that $[[a_1,a_2,\cdots ,a_{n+1}]]$ is $0$-clean iff $(a_1,a_2,\cdots ,a_{n+1})=1$. As an application, we prove that the $0$-cleanness and unit-regularity for such $n\times n$ matrix over a Dedekind domain coincide for all $n\geq 3$. The analogous for $(s,2)$ property is also obtained.

Prestability for quadratic [formula omitted] of Λ-1-fold stable rings

The general quadratic group U 2 n and its elementary subgroup EU 2 n are analogs in the theory of the general linear group GL n and its elementary subgroup E n . It is known, for GL n , K 1 = GL 1 / 〈 ( a + c + a b c ) ( a + c + c b a ) −1 | a + c + a b c ∈ GL 1 〉 under the first Bass stable range condition on the ring of entries. In this article we give a description for KU 1 under the Λ-1-fold stable range condition on the ring of entries.

Unit-Regularity and Stable Range Conditions#

ABSTRACT A ring R has stable range one (respectively, weakly stable range one) provided whenever ax + b = 1 in R , there exists y ∈ R such that a + by is a two-sided (respectively, one-sided) inverse. In the first part of this article, we show that an exchange ring R has stable range one if and only if for every regular element x ∈ R there exist some unit-regular element w ∈ R and some idempotent e ∈ R such that x = ew if and only if whenever aR + bR = dR in R , there exists a unit-regular element w ∈ R such that a + bz = dw for some z ∈ R . These generalize a result of Canfell (1995, Theorem 2.9) and a special case of Chen (2000b, Theorem 4.2). In the second part, we prove that a module M satisfies the internal weak cancellation property if and only if every regular element of the endomorphism ring of M is one-sided unit-regular. We also extend a well known result on exchange rings with stable range one to exchange rings with weakly stable range one, by showing that an exchange ring R has weakly stable range one if and only if whenever in R , there exists a one-sided inverse u ∈ R such that au = ub .

Extensions of Unit 1-Stable Range#

In this paper, we investigate necessary and sufficient conditions under which a ring R satisfies unit ideal–stable range conditions. We prove that if I is a bounded regular ideal of a ring R, then unit (I)-stable range condition, unit ⟨I⟩-stable range condition and unit [I]-stable range condition coincide with each other.

Rings with internal cancellation

In this paper, we study the class of rings that satisfy internal direct sum cancellation with respect to their 1-sided ideals. These are known to be precisely the rings in which regular elements are unit-regular. Further characterizations for such “IC rings” are given, in terms of suitable versions of stable range conditions, and unique generator properties of idempotent generated right ideals. This approach leads to a uniform treatment of many of the known characterizations for an exchange ring to have stable range 1. Rings whose matrix rings are IC turn out to be precisely those rings whose finitely generated projective modules satisfy cancellation. We also offer a couple of “hidden” characterizations of unit-regular elements in rings that shed some new light on the relation between similarity and pseudo-similarity—in monoids as well as in rings. The paper concludes with a treatment of ideals for which idempotents lift modulo all 1-sided subideals. An appendix by R.G. Swan 1 1 Address: 700 Melrose Av., Apt. M3, Winter Park, FL 32789, USA. E-mail address: swan@math.uchicago.edu. on the failure of cancellation for finitely generated projective modules over complex group algebras shows that such algebras are in general not IC.

Extensions of Rings with Many Units

In this paper, we investigate stable range conditions over extensions of matrix rings. It is shown that a ring R satisfies (s, 2)-stable range if and only if R has a complete orthogonal set {e 1,…, e n } of idempotents such that all e i Re i satisfy (s, 2)-stable range. Also we extend this result to (s, 2)-rings and rings satisfying unit 1-stable range.

Projection geometry over rings

A generalization of the fundamental theorem of projective geometry is established for non-injective mappings between projective geometries defined over rings which satisfy a stable range condition. A geometrical characterization of the stable range condition is also given.

MORITA CONTEXTS WITH MANY UNITS

Abstract In this paper, we investigate stable range conditions over Morita contexts. As applications, we calculate the Whitehead groups of Morita contexts for exchange rings with primitive factors artinian.

Morita Contexts with Many Units

In this paper, we investigate stable range conditions over Morita contexts. As applications, we calculate the Whitehead groups of Morita contexts for exchange rings with primitive factors artinian.

Stable Ranges For Morita Contexts

In this paper, we show that if rings A and B are (s, 2)-rings, then so is the ring of a Morita context (A, B, M, N, ψ, φ). Also we get analogous results for unit 1-stable ranges and GM-rings. These give new classes of rings satisfying such stable range conditions.

An Infinite Analogue of Rings with Stable Rank One

Replacing invertibility with quasi-invertibility in Bass' first stable range condition we discover a new class of rings, the QB-rings. These constitute a considerable enlargement of the class of rings with stable rank one (B-rings) and include examples like EndF(V), the ring of endomorphisms of a vector space V over some field F, and B(F), the ring of all row- and column-finite matrices over F. We show that the category of QB-rings is stable under the formation of corners, ideals, and quotients, as well as matrices and direct limits. We also give necessary and sufficient conditions for an extension of QB-rings to be a QB-ring, and show that extensions of B-rings often lead to QB-rings. Specializing to the category of exchange rings we characterize the subset of exchange QB-rings as those in which every von Neumann regular element extends to a maximal regular element, i.e., a quasi-invertible element. Finally we show that the C*-algebras that are QB-rings are exactly the extremally rich C*-algebras studied by L. G. Brown and the second author.