Exchange rings having ideal-stable range one

In this paper, we introduce the notion of the ideal-stable range one condition for exchange rings. Some characterizations for this condition are given. Moreover, we show that, for an exchange ringR, ifI is an ideal ofR andR hasI-stable range one, then every regular square matrix overI is the product of an idempotent matrix and an invertible matrix overR, and admits a diagonal reduction.     

Stable range one for rings with central units

Abstract

The purpose of this paper is to give a partial positive answer to a question raised by Khurana et al. as to whether a ring R with stable range one and central units is commutative. We show that this is the case under any of the following additional conditions: R is semiprime or R is one-sided Noetherian or R has unit-stable range 1 or R has classical Krull dimension 0 or R is an algebra over a field K such that K is uncountable and R has only countably many primitive ideals or R is affine and either K has characteristic 0 or has infinite transcendental degree over its prime subfield or is algebraically closed. However, the general question remains open.     

Exchange rings in which all regular elements are one-sided unit-regular

Let R be an exchange ring in which all regular elements are one-sided unit-regular. Then every regular element in R is the sum of an idempotent and a one-sided unit. Furthermore, we extend this result to exchange rings satisfying related comparability.     

Stable subsets of modules and the existence of a unit in associative rings

We obtain criteria for the existence of a (left) unit in rings (arbitrary, Artinian, Noetherian, prime, and so on) that are based on the systematic study of properties of stable subsets of modules and their stabilizers that generalize the technique of idempotents. We study a class of quasiunitary rings that is a natural extension of classes of rings with unit and of von Neumann (weakly) regular rings, which inherits may properties of these classes. Some quasiunitary radicals of arbitrary rings are constructed, and the size of these radicals “measures the probability” of the existence of a unit. Translated fromMatematicheskie Zametki, Vol. 61, No. 4, pp. 596–611, April, 1997. Translated by A. I. Shtern     

Elementary matrix reduction over certain rings

We explore elementary matrix reduction over certain rings characterized by properties related to stable range. Let R be a commutative ring. We call R locally stable if aR+bR = R??x∈R such that R∕(a+bx)R has stable range 1. We study locally stable rings and prove that every locally stable Bézout ring is an elementary divisor ring. Many known results on domains are thereby generalized.     

Exchange rings satisfying power ideal-cancellation

In this paper, we introduce a new kind of partial power cancellations, and get a number of necessary and sufficient conditions for exchange rings satisfying power ideal-cancellation. These also extend many known results.     

Generalized Stable Exchange Rings

In this paper, we investigate diagonal reductions of matrices over generalized stable exchange rings. We show that every regular matirx over generalized stable exchange rings with stable range 2 admits power diagonal reduction.AMS Subject Classification (1991): 16A30 16E50This work was supported by the National Natural Science Foundation of China.     

置换环上的稳定矩阵

本文证明了置换环上的正则稳定矩阵是幂等矩阵和可逆矩阵的积,进一步证明了置换环上的正则稳定矩阵可以对角化。     

Prime ideals of principally quasi-Baer rings

We extend a theorem of Kist for commutative PP rings to principally quasi-Baer rings for which every prime ideal contains a unique minimal prime ideal without using topological arguments. Also decompositions of quasi-Baer and principally quasi-Baer rings are investigated. This revised version was published online in June 2006 with corrections to the Cover Date.     

Principal quasi-Baerness of formal power series rings

Let R be a ring. We consider left （or right） principal quasi-Baerness of the left skew formal power series ring R[[x;α]] over R where a is a ring automorphism of R. We give a necessary and sufficient condition under which the ring R[[x; α]] is left （or right） principally quasi-Baer. As an application we show that R[[x]] is left principally quasi-Baer if and only if R is left principally quasi- Baer and the left annihilator of the left ideal generated by any countable family of idempotents in R is generated by an idempotent.     

On the isomorphisms of the character rings

By considering the Galois action and the group action,we give the relationof two isomorphism character rings on the arbitrary field K with Char K=0.Our resultsgeneralize Saksonov's theorem.     

Projective geometry over rings with stable range condition

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.     

单边EXCHANGE环

本文引进研究了单边Exchange环和Msta-sjded Exchange环．给出了单边Exchange环的一类等价条件，得到了约化条件下这几类环的等价性．证明了单边EXCHANGE环上模的直和消去也等价于部分单位正则性．     

Quasi-duo rings and stable range descent

In a recent paper, the first author introduced a general theory of corner rings in noncommutative rings that generalized the classical theory of Peirce decompositions. This theory is applied here to the study of the stable range of rings upon descent to corner rings. A ring is called quasi-duo if every maximal 1-sided ideal is 2-sided. Various new characterizations are obtained for such rings. Using some of these characterizations, we prove that, if a quasi-duo ring R has stable range ?n, the same is true for any semisplit corner ring of R. This contrasts with earlier results of Vaserstein and Warfield, which showed that the stable range can increase unboundedly upon descent to (even) Peirce corner rings.     

Strong separativity over exchange rings

An exchange ring R is strongly separative provided that for all finitely generated projective right R-modules A and B, A ⊕ A ≅ A ⊕ B ⇒ A ≅ B. We prove that an exchange ring R is strongly separative if and only if for any corner S of R, aS + bS = S implies that there exist u, v ∈ S such that au = bv and Su + Sv = S if and only if for any corner S of R, aS + bS = S implies that there exists a right invertible matrix ∈ M ₂(S). The dual assertions are also proved.     

Polar spaces over rings with absolute stable range condition

A generalization of the fundamental theorem of projective geometry is established for non-injective mappings between polar spaces defined over form rings which satisfy a stable range condition.     

Bézout rings with almost stable range 1

Elementary divisor domains were defined by Kaplansky [I. Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc. 66 (1949) 464-491] and generalized to rings with zero-divisors by Gillman and Henriksen [L. Gillman, M. Henriksen, Some remarks about elementary divisor rings, Trans. Amer. Math. Soc. 82 (1956) 362-365]. In [M.D. Larsen, W.J. Lewis, T.S. Shores, Elementary divisor rings and finitely presented modules, Trans. Amer. Math. Soc. 187 (1) (1974) 231-248], it was also proved that if a Hermite ring satisfies (N), then it is an elementary divisor ring. The aim of this article is to generalize this result (as well as others) to a much wider class of rings. Our main result is that Bézout rings whose proper homomorphic images all have stable range 1 (in particular, neat rings) are elementary divisor rings.