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 Rings in which elements are the sum of a nilpotent and a root of a fixed polynomial that commute

2017 ◽  
Vol 15 (1) ◽  
pp. 420-426 ◽  
Author(s):  
Ali H. Handam ◽  
Hani A. Khashan
Keyword(s):  
Basic Properties ◽  
Clean Rings

Abstract An element in a ring R with identity is said to be strongly nil clean if it is the sum of an idempotent and a nilpotent that commute, R is said to be strongly nil clean if every element of R is strongly nil clean. Let C(R) be the center of a ring R and g(x) be a fixed polynomial in C(R)[x]. Then R is said to be strongly g(x)-nil clean if every element in R is a sum of a nilpotent and a root of g(x) that commute. In this paper, we give some relations between strongly nil clean rings and strongly g(x)-nil clean rings. Various basic properties of strongly g(x) -nil cleans are proved and many examples are given.

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 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.

scholarly journals On γ-Clean Rings

2017 ◽  
Vol 5 (3) ◽  
pp. 285
Author(s):  
Shaimaa S. Esa ◽  
Hewa S. Faris
Keyword(s):  
Basic Properties ◽  
Clean Rings ◽  
Clean Ring

In this paper we introduce the concept of -clean ring and we discuss some relations between - clean ring and other rings with explaining by some examples. Also, we give some basic properties of it.

USING IDEALS TO PROVIDE A UNIFIED APPROACH TO UNIQUELY CLEAN RINGS

2013 ◽  
Vol 96 (2) ◽  
pp. 258-274
Author(s):  
V. A. HIREMATH ◽  
SHARAD HEGDE
Keyword(s):  
Prime Ideal ◽  
Ideal Extension ◽  
Central Idempotent ◽  
Exchange Ring ◽  
Unified Approach ◽  
Clean Rings ◽  
Clean Ring ◽  
Nil Clean Ring

AbstractIn this article, we introduce the notion of the uniquely $I$-clean ring and show that, if $R$ is a ring and $I$ is an ideal of $R$ then $R$ is uniquely $I$-clean if and only if ($R/ I$ is Boolean and idempotents lift uniquely modulo $I$) if and only if (for each $a\in R$ there exists a central idempotent $e\in R$ such that $e- a\in I$ and $I$ is idempotent-free). We examine when ideal extension is uniquely clean relative to an ideal. Also we obtain conditions on a ring $R$ and an ideal $I$ of $R$ under which uniquely $I$-clean rings coincide with uniquely clean rings. Further we prove that a ring $R$ is uniquely nil-clean if and only if ($N(R)$ is an ideal of $R$ and $R$ is uniquely $N(R)$-clean) if and only if $R$ is both uniquely clean and nil-clean if and only if ($R$ is an abelian exchange ring with $J(R)$ nil and every quasiregular element is uniquely clean). We also show that $R$ is a uniquely clean ring such that every prime ideal of $R$ is maximal if and only if $R$ is uniquely nil-clean ring and $N(R)= {\mathrm{Nil} }_{\ast } (R)$.

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.

Exchange rings, exchange equations, and lifting properties

2016 ◽  
Vol 26 (06) ◽  
pp. 1177-1198 ◽  
Author(s):  
Dinesh Khurana ◽  
T. Y. Lam ◽  
Pace P. Nielsen
Keyword(s):  
Commutative Rings ◽  
Von Neumann ◽  
Square Roots ◽  
New Class ◽  
Clean Rings ◽  
Von Neumann Regular Rings ◽  
Exchange Rings ◽  
Von Neumann Regular ◽  
Regular Rings

In this paper, we study exchange rings and clean rings [Formula: see text] with [Formula: see text] (or otherwise). Analogues of a theorem of Camillo and Yu characterizing clean and strongly clean rings with [Formula: see text] are obtained for such rings (as well as for exchange rings) using the viewpoint of exchange equations introduced in a recent paper of the authors. We also study a new class of rings including von Neumann regular rings in which square roots of one (instead of idempotents) can be lifted modulo left ideals, and conjecture that such rings are exchange rings. This conjecture holds for commutative rings, and would hold for all rings if it holds for semiprimitive rings of characteristic [Formula: see text].

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