scholarly journals Elements in exchange rings with related comparability

2000 ◽  
Vol 23 (9) ◽  
pp. 639-644 ◽  
Author(s):  
Huanyin Chen
Keyword(s):  
Exchange Ring ◽  
Dual Problems ◽  
Exchange Rings ◽  
Related Comparability

We show that ifRis an exchange ring, then the following are equivalent: (1)Rsatisfies related comparability. (2) Givena,b,d∈RwithaR+bR=dR, there exists a related unitw∈Rsuch thata+bt=dw. (3) Givena,b∈RwithaR=bR, there exists a related unitw∈Rsuch thata=bw. Moreover, we investigate the dual problems for rings which are quasi-injective as right modules.

EXCHANGE RINGS OVER WHICH EVERY REGULAR MATRIX ADMITS DIAGONAL REDUCTION

2004 ◽  
Vol 03 (02) ◽  
pp. 207-217 ◽  
Author(s):  
HUANYIN CHEN
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Exchange Ring ◽  
Regular Matrix ◽  
Finitely Generated ◽  
Exchange Rings ◽  
Necessary And Sufficient ◽  
Diagonal Reduction

In this paper, we investigate the necessary and sufficient conditions on exchange rings, under which every regular matrix admits diagonal reduction. Also we show that an exchange ring R is strongly separative if and only if for any finitely generated projective right R-module C, if A and B are any right R-modules such that 2C⊕A≅C⊕B, then C⊕A≅B.

On Exchange Hermitian Rings

2010 ◽  
Vol 17 (01) ◽  
pp. 87-100 ◽  
Author(s):  
Huanyin Chen
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Exchange Ring ◽  
Regular Matrix ◽  
Exchange Rings ◽  
Necessary And Sufficient ◽  
Diagonal Reduction

In this article, we investigate new necessary and sufficient conditions on an exchange ring under which every regular matrix admits a diagonal reduction. We prove that an exchange ring R is an hermitian ring if and only if for any n ≥ 2 and any regular x ∈ Rn, there exists u ∈ CLn(R) such that x = xux; if and only if for any n ≥ 2 and any regular x ∈ Rn, there exists u ∈ CLn(R) such that xu ∈ R is an idempotent. Furthermore, we characterize such exchange rings by means of reflexive inverses and n-pseudo-similarity.

scholarly journals Exchange rings having stable range one

2001 ◽  
Vol 25 (12) ◽  
pp. 763-770 ◽  
Author(s):  
Huanyin Chen
Keyword(s):  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Stable Range ◽  
Exchange Ring ◽  
Stable Range One ◽  
Exchange Rings

We investigate the sufficient conditions and the necessary conditions on an exchange ringRunder whichRhas stable range one. These give nontrivial generalizations of Theorem 3 of V. P. Camillo and H.-P. Yu (1995), Theorem 4.19 of K. R. Goodearl (1979, 1991), Theorem 2 of R. E. Hartwig (1982), and Theorem 9 of H.-P. Yu (1995).

scholarly journals A CLASS OF EXCHANGE RINGS

2008 ◽  
Vol 50 (3) ◽  
pp. 509-522 ◽  
Author(s):  
TSIU-KWEN LEE ◽  
YIQIANG ZHOU
Keyword(s):  
Exchange Ring ◽  
Glasgow Math ◽  
Basic Properties ◽  
Boolean Rings ◽  
Clean Rings ◽  
Exchange Rings

AbstractIt is well known that a ring R is an exchange ring iff, for any a ∈ R, a−e ∈ (a2−a)R for some e2 = e ∈ R iff, for any a ∈ R, a−e ∈ R(a2−a) for some e2 = e ∈ R. The paper is devoted to a study of the rings R satisfying the condition that for each a ∈ R, a−e ∈ (a2−a)R for a unique e2 = e ∈ R. This condition is not left–right symmetric. The uniquely clean rings discussed in (W. K. Nicholson and Y. Zhou, Rings in which elements are uniquely the sum of an idempotent and a unit, Glasgow Math. J. 46 (2004), 227–236) satisfy this condition. These rings are characterized as the semi-boolean rings with a restricted commutativity for idempotents, where a ring R is semi-boolean iff R/J(R) is boolean and idempotents lift modulo J(R) (or equivalently, R is an exchange ring for which any non-zero idempotent is not the sum of two units). Various basic properties of these rings are developed, and a number of illustrative examples are given.

scholarly journals On some constructions of nil-clean, clean and exchange rings

2015 ◽  
Vol 14 (07) ◽  
pp. 1550101
Author(s):  
Alin Stancu
Keyword(s):  
Jacobson Radical ◽  
Exchange Ring ◽  
Common Theme ◽  
Infinitesimal Deformation ◽  
Formal Deformation ◽  
Exchange Rings ◽  
Infinitesimal Deformations ◽  
The Common

In this paper, we discuss several constructions that lead to new examples of nil-clean, clean and exchange rings. Extensions by ideals contained in the Jacobson radical is the common theme of these constructions. A characterization of the idempotents in the algebra defined by a 2-cocycle is given and used to prove some of the algebra's properties (the infinitesimal deformation case). From infinitesimal deformations, we go to full deformations and prove that any formal deformation of a clean (exchange) ring is itself clean (exchange). Examples of nil-clean, clean and exchange rings, arising from poset algebras are also discussed.

Generalized Stable Exchange Rings

2008 ◽  
Vol 15 (02) ◽  
pp. 193-198 ◽  
Author(s):  
Huanyin Chen
Keyword(s):  
Exchange Ring ◽  
Regular Matrix ◽  
Exchange Rings ◽  
Invertible Matrices ◽  
Diagonal Reduction ◽  
Stable Ring

A ring R is said to be a generalized stable ring provided that aR + bR = R with a, b ∈ R implies that there exists y ∈ R such that a + by ∈ K(R), where K(R) = {x ∈ R | ∃s, t ∈ R such that sxt = 1}. Let A be a quasi-projective right R-module, and let E = End R(A). If E is an exchange ring, then E is a generalized stable ring if and only if for any R-morphism f : A → M with Im f ≤⊕M and any R-epimorphism g : A → M, there exist e = e2 ∈ E and h ∈ K(E) such that f = g(eh). Furthermore, we prove that every regular matrix over a generalized stable exchange ring admits a diagonal reduction by quasi-invertible matrices.

Related comparability over exchange rings

1999 ◽  
Vol 27 (9) ◽  
pp. 4209-4216 ◽  
Author(s):  
Huanyin Chen
Keyword(s):  
Exchange Rings ◽  
Related Comparability

SPLIT-EXTENSIONS OF EXCHANGE RINGS: A DIRECT PROOF

2009 ◽  
Vol 08 (05) ◽  
pp. 629-632
Author(s):  
GRIGORE CĂLUGĂREANU
Keyword(s):  
Direct Proof ◽  
Exchange Ring ◽  
Exchange Rings ◽  
Theoretic Proof ◽  
Split Extensions

A direct ring-theoretic proof for "if R is a ring, e is an idempotent and eRe and (1 - e)R(1 - e) are both exchange rings then R is also an exchange ring" (left open for the last 31 years) is given.

scholarly journals Whitehead groups of exchange rings with primitive factors artinian

2001 ◽  
Vol 26 (7) ◽  
pp. 393-398 ◽  
Author(s):  
Huanyin Chen ◽  
Fu-an Li
Keyword(s):  
Exchange Ring ◽  
Group Of Units ◽  
Exchange Rings ◽  
Whitehead Groups

We show that ifRis an exchange ring with primitive factors artinian thenK1(R)≅U(R)/V(R), whereU(R)is the group of units ofRandV(R)is the subgroup generated by{(1+ab)(1+ba)−1|a,b∈R   with   1+ab∈U(R)}. As a corollary,K1(R)is the abelianized group of units ofRif1/2∈R.

Sign in / Sign up

Export Citation Format

Share Document