STRONGLY CLEAN MATRICES OVER COMMUTATIVE LOCAL RINGS

2012 ◽  
Vol 12 (01) ◽  
pp. 1250126 ◽  
Author(s):  
H. CHEN ◽  
O. GURGUN ◽  
H. KOSE
Keyword(s):  
University Of California ◽  
Local Rings ◽  
Matrix Rings ◽  
Algebraic Analysis ◽  
Clean Rings ◽  
Memorial University Of Newfoundland

An element of a ring is called strongly clean provided that it can be written as the sum of an idempotent and a unit that commute. We characterize, in this paper, the strongly cleanness of matrices over commutative local rings. This partially extend many known results such as Theorem 12 in Borooah, Diesl and Dorsey [Strongly clean matrix rings over commutative local rings, J. Pure Appl. Algebra212 (2008) 281–296], Theorem 3.2.7 and Proposition 3.3.6 in Dorsey [Cleanness and strong cleanness of rings of matrices, Ph.D. thesis, University of California, Berkeley (2006)], Theorem 2.3.14 in Fan [Algebraic analysis of some strongly clean and their generalization, Ph.D. thesis, Memorial University of Newfoundland, Newfoundland (2009)], Theorem 3.1.9 and Theorem 3.1.26 in Yang [Strongly clean rings and g(x)-clean rings, Ph.D. thesis, Memorial University of Newfoundland, Newfoundland (2007)].

PSEUDOPOLAR MATRIX RINGS OVER LOCAL RINGS

2013 ◽  
Vol 13 (03) ◽  
pp. 1350109 ◽  
Author(s):  
JIAN CUI ◽  
JIANLONG CHEN
Keyword(s):  
Positive Integer ◽  
Local Rings ◽  
Matrix Rings ◽  
Clean Rings ◽  
Regular Rings

A ring R is called pseudopolar if for every a ∈ R there exists p2 = p ∈ R such that p ∈ comm 2(a), a + p ∈ U(R) and akp ∈ J(R) for some positive integer k. Pseudopolar rings are closely related to strongly π-regular rings, uniquely strongly clean rings, semiregular rings and strongly π-rad clean rings. In this paper, we completely characterize the local rings R for which M2(R) is pseudopolar.

Commutative nil clean group rings

2015 ◽  
Vol 14 (06) ◽  
pp. 1550094 ◽  
Author(s):  
Warren Wm. McGovern ◽  
Shan Raja ◽  
Alden Sharp
Keyword(s):  
Commutative Group ◽  
University Of California ◽  
Short Paper ◽  
Group Rings ◽  
Clean Rings ◽  
Clean Ring ◽  
Nil Clean Ring

In [A. J. Diesl, Classes of strongly clean rings, Ph.D. Dissertation, University of California, Berkely (2006); Nil clean rings, J. Algebra383 (2013) 197–211], a nil clean ring was defined as a ring for which every element is the sum of a nilpotent and an idempotent. In this short paper, we characterize nil clean commutative group rings.

Nil-Clean Rings with Involution

2021 ◽  
Vol 28 (03) ◽  
pp. 367-378
Author(s):  
Jian Cui ◽  
Guoli Xia ◽  
Yiqiang Zhou
Keyword(s):  
Boolean Ring ◽  
Matrix Rings ◽  
Clean Rings ◽  
Clean Ring ◽  
Rings With Involution ◽  
Text Version

A [Formula: see text]-ring [Formula: see text] is called a nil [Formula: see text]-clean ring if every element of [Formula: see text] is a sum of a projection and a nilpotent. Nil [Formula: see text]-clean rings are the [Formula: see text]-version of nil-clean rings introduced by Diesl. This paper is about the nil [Formula: see text]-clean property of rings with emphasis on matrix rings. We show that a [Formula: see text]-ring [Formula: see text] is nil [Formula: see text]-clean if and only if [Formula: see text] is nil and [Formula: see text] is nil [Formula: see text]-clean. For a 2-primal [Formula: see text]-ring [Formula: see text], with the induced involution given by[Formula: see text], the nil [Formula: see text]-clean property of [Formula: see text] is completely reduced to that of [Formula: see text]. Consequently, [Formula: see text] is not a nil [Formula: see text]-clean ring for [Formula: see text], and [Formula: see text] is a nil [Formula: see text]-clean ring if and only if [Formula: see text] is nil, [Formula: see text]is a Boolean ring and [Formula: see text] for all [Formula: see text].

scholarly journals CS matrix rings over local rings

2003 ◽  
Vol 264 (1) ◽  
pp. 251-261 ◽  
Author(s):  
K.I. Beidar ◽  
S.K. Jain ◽  
Pramod Kanwar ◽  
J.B. Srivastava
Keyword(s):  
Local Rings ◽  
Matrix Rings

FACTOR RINGS OF PSEUDO-FROBENIUS RINGS

2006 ◽  
Vol 05 (06) ◽  
pp. 847-854 ◽  
Author(s):  
CARL FAITH
Keyword(s):  
Left Ideal ◽  
Single Element ◽  
Local Rings ◽  
Central Idempotent ◽  
Main Result Theorem ◽  
Matrix Rings ◽  
Frobenius Rings ◽  
Result Theorem ◽  
Finite Products ◽  
Left Annihilator

If R is right pseudo-Frobenius (= PF), and A is an ideal, when is R/A right PF? Our main result, Theorem 3.7, states that this happens iff the ideal A′ of the basic ring B of R corresponding to A has left annihilator F in B generated by a single element on both sides. Moreover, in this case B/A′ ≈ F in mod-B, (see Theorem 3.5), a property that does not extend to R, that is, in general R/A is not isomorphic to the left annihilator of A. (See Example 4.3(2) and Theorem 4.5.) Theorem 4.6 characterizes Frobenius rings among quasi-Frobenius (QF) rings. As an application of the main theorem, in Theorem 3.9 we prove that if A is generated as a right or left ideal by an idempotent e, then e is central (and R/A is then trivially right PF along with R). This generalizes the result of F. W. Anderson for quasi-Frobenius rings. (See Theorem 2.2 for a new proof.). In Proposition 1.6, we prove that a generalization of this result holds for finite products R of full matrix rings over local rings; namely, an ideal A is finitely generated as a right or left ideal iff A is generated by a central idempotent. We also note a theorem going back to Nakayama, Goursaud, and the author that every factor ring of R is right PF iff R is a uniserial ring. (See Theorem 5.1.).

scholarly journals Isomorphism of generalized triangular matrix-rings and recovery of tiles

2003 ◽  
Vol 2003 (9) ◽  
pp. 533-538 ◽  
Author(s):  
R. Khazal ◽  
S. Dăscălescu ◽  
L. Van Wyk
Keyword(s):  
Matrix Ring ◽  
Triangular Matrix ◽  
Commutative Rings ◽  
Local Rings ◽  
Isomorphism Theorem ◽  
Matrix Rings ◽  
Recovery Result

We prove an isomorphism theorem for generalized triangular matrix-rings, over rings having only the idempotents0and1, in particular, over indecomposable commutative rings or over local rings (not necessarily commutative). As a consequence, we obtain a recovery result for the tile in a tiled matrix-ring.

scholarly journals Strongly clean matrix rings over local rings

2007 ◽  
Vol 312 (1) ◽  
pp. 397-404 ◽  
Author(s):  
Yuanlin Li
Keyword(s):  
Local Rings ◽  
Matrix Rings

Strongly unit nil-clean rings

2017 ◽  
Vol 16 (06) ◽  
pp. 1750115
Author(s):  
Huanyin Chen ◽  
Marjan Sheibani
Keyword(s):  
Nilpotent Element ◽  
Matrix Rings ◽  
Clean Rings ◽  
Clean Element ◽  
Unit Formula

An element in a ring is strongly nil-clean, if it is the sum of an idempotent and a nilpotent element that commute. A ring [Formula: see text] is strongly unit nil-clean, if for any [Formula: see text] there exists a unit [Formula: see text], such that [Formula: see text] is strongly nil-clean. We prove, in this paper, that a ring [Formula: see text] is strongly unit nil-clean, if and only if every element in [Formula: see text] is equivalent to a strongly nil-clean element, if and only if for any [Formula: see text], there exists a unit [Formula: see text], such that [Formula: see text] is strongly [Formula: see text]-regular. Strongly unit nil-clean matrix rings are investigated as well.

scholarly journals Some Remarks on Non-Commutative Extensions of Local Rings

1959 ◽  
Vol 14 ◽  
pp. 45-51
Author(s):  
Edward H. Batho
Keyword(s):  
Local Ring ◽  
Ideal Theory ◽  
Dimension Theory ◽  
Local Rings ◽  
Important Class ◽  
Matrix Rings ◽  
Commutative Local Ring

In [1] we introduced the concept of a non-commutative local ring and studied the structure of such rings. Unfortunately, we were not able to show that the completion of a local ring was a semi-local ring. In this paper we propose to study a class of rings for which the above result is valid. This class of rings is the integral extensions -[4, 5]-of commutative local rings. This class of rings includes the important class of matrix rings over commutative local rings. In part 1 below we study some elementary properties of integral extensions and here we assume merely that the underlying ring is semi-local. In part 2 we discuss some questions of ideal theory for arbitrary local rings as well as for integral extensions. In a later paper we propose to utilize our results to study the deeper properties of these rings including a dimension theory for such rings. We are particularly indebted to the work of Nagata [6, 7, 8, 9] in the preparation of this paper.

Sign in / Sign up

Export Citation Format

Share Document