scholarly journals On two open problems about strongly clean rings

2004 ◽  
Vol 70 (2) ◽  
pp. 279-282 ◽  
Author(s):  
Zhou Wang ◽  
Jianlong Chen
Keyword(s):  
Matrix Ring ◽  
Triangular Matrix ◽  
Open Problems ◽  
Clean Rings ◽  
Clean Ring ◽  
The Matrix ◽  
Triangular Matrix Ring ◽  
Semiperfect Ring ◽  
Upper Triangular Matrix ◽  
Upper Triangular Matrix Ring

A ring is called strongly clean if every element is the sum of an idempotent and a unit which commute. In 1999 Nicholson asked whether every semiperfect ring is strongly clean and whether the matrix ring of a strongly clean ring is strongly clean. In this paper, we prove that if R = {m/n ∈ ℚ: n is odd}, then M2(R) is a semiperfect ring but not strongly clean. Thus, we give negative answers to both questions. It is also proved that every upper triangular matrix ring over the ring R is strongly clean.

scholarly journals Abel Rings and Super-Strongly Clean Rings

2015 ◽  
Vol 0 (0) ◽  
Author(s):  
Yinchun Qu ◽  
Junchao Wei
Keyword(s):  
Matrix Ring ◽  
Triangular Matrix ◽  
Clean Rings ◽  
Clean Ring ◽  
Triangular Matrix Ring ◽  
Upper Triangular Matrix ◽  
Upper Triangular Matrix Ring

Abstract In this note, we first show that a ring R is Abel if and only if the 2 × 2 upper triangular matrix ring over R is quasi-normal. Next, we give the notion of super-strongly clean ring (that is, an Abel clean ring), which is inbetween uniquely clean rings and strongly clean rings. Some characterizations of super-strongly clean rings are given.

On Quasipolar Rings

2012 ◽  
Vol 19 (04) ◽  
pp. 683-692 ◽  
Author(s):  
Zhiling Ying ◽  
Jianlong Chen
Keyword(s):  
Matrix Ring ◽  
Triangular Matrix ◽  
Commutative Rings ◽  
Ring Theory ◽  
Clean Rings ◽  
Commutative Local Ring ◽  
Triangular Matrix Ring ◽  
Upper Triangular Matrix ◽  
Upper Triangular Matrix Ring ◽  
Regular Rings

The notion of quasipolar elements of rings was introduced by Koliha and Patricio in 2002. In this paper, we introduce the notion of quasipolar rings and relate it to other familiar notions in ring theory. It is proved that both strongly π-regular rings and uniquely clean rings are quasipolar, and quasipolar rings are strongly clean, but no two of these classes of rings are equivalent. For commutative rings, quasipolar rings coincide with semiregular rings. It is also proved that every n × n upper triangular matrix ring over any commutative uniquely clean ring or commutative local ring is quasipolar.

Integer-valued polynomials over block matrix algebras

2019 ◽  
Vol 19 (03) ◽  
pp. 2050053
Author(s):  
J. Sedighi Hafshejani ◽  
A. R. Naghipour ◽  
M. R. Rismanchian
Keyword(s):  
Matrix Ring ◽  
Triangular Matrix ◽  
Block Matrix ◽  
Quotient Field ◽  
Matrix Rings ◽  
Matrix Algebras ◽  
Triangular Matrix Ring ◽  
Upper Triangular Matrix ◽  
Full Matrix Ring ◽  
Upper Triangular Matrix Ring

In this paper, we state a generalization of the ring of integer-valued polynomials over upper triangular matrix rings. The set of integer-valued polynomials over some block matrix rings is studied. In fact, we consider the set of integer-valued polynomials [Formula: see text] for each [Formula: see text], where [Formula: see text] is an integral domain with quotient field [Formula: see text] and [Formula: see text] is a block matrix ring between upper triangular matrix ring [Formula: see text] and full matrix ring [Formula: see text]. In fact, we have [Formula: see text]. It is known that the sets of integer-valued polynomials [Formula: see text] and [Formula: see text] are rings. We state some relations between the rings [Formula: see text] and the partitions of [Formula: see text]. Then, we show that the set [Formula: see text] is a ring for each [Formula: see text]. Further, it is proved that if the ring [Formula: see text] is not Noetherian then the ring [Formula: see text] is not Noetherian, too. Finally, some properties and relations are stated between the rings [Formula: see text], [Formula: see text] and [Formula: see text].

Projection invariant extending rings

2016 ◽  
Vol 15 (07) ◽  
pp. 1650121 ◽  
Author(s):  
Gary F. Birkenmeier ◽  
Adnan Tercan ◽  
Canan C. Yucel
Keyword(s):  
Direct Summand ◽  
Matrix Ring ◽  
Triangular Matrix ◽  
Trivial Extension ◽  
Matrix Rings ◽  
Finite Matrix ◽  
Triangular Matrix Ring ◽  
Upper Triangular Matrix ◽  
The Right ◽  
Upper Triangular Matrix Ring

A ring [Formula: see text] is said to be right [Formula: see text]-extending if every projection invariant right ideal of [Formula: see text] is essential in a direct summand of [Formula: see text]. In this article, we investigate the transfer of the [Formula: see text]-extending condition between a ring [Formula: see text] and its various ring extensions. More specifically, we characterize the right [Formula: see text]-extending generalized triangular matrix rings; and we show that if [Formula: see text] is [Formula: see text]-extending, then so is [Formula: see text] where [Formula: see text] is an overring of [Formula: see text] which is an essential extension of [Formula: see text], an [Formula: see text] upper triangular matrix ring of [Formula: see text], a column finite or column and row finite matrix ring over [Formula: see text], or a certain type of trivial extension of [Formula: see text].

scholarly journals ON WEAKLY RIGID RINGS

2009 ◽  
Vol 51 (3) ◽  
pp. 425-440 ◽  
Author(s):  
A. R. NASR-ISFAHANI ◽  
A. MOUSSAVI
Keyword(s):  
Prime Ring ◽  
Matrix Ring ◽  
Triangular Matrix ◽  
The Other ◽  
Rigid Ring ◽  
Triangular Matrix Ring ◽  
Upper Triangular Matrix ◽  
Upper Triangular Matrix Ring

AbstractLet R be a ring with a monomorphism α and an α-derivation δ. We introduce (α, δ)-weakly rigid rings which are a generalisation of α-rigid rings and investigate their properties. Every prime ring R is (α, δ)-weakly rigid for any automorphism α and α-derivation δ. It is proved that for any n, a ring R is (α, δ)-weakly rigid if and only if the n-by-n upper triangular matrix ring Tn(R) is (, )-weakly rigid if and only if Mn(R) is (, )-weakly rigid. Moreover, various classes of (α, δ)-weakly rigid rings is constructed, and several known results are extended. We show that for an (α, δ)-weakly rigid ring R, and the extensions R[x], R[[x]], R[x; α, δ], R[x, x−1; α], R[[x; α]], R[[x, x−1; α]], the ring R is quasi-Baer if and only if the extension over R is quasi-Baer. It is also proved that for an (α, δ)-weakly rigid ring R, if any one of the rings R, R[x], R[x; α, δ] and R[x, x−1; α] is left principally quasi-Baer, then so are the other three. Examples to illustrate and delimit the theory are provided.

Rings in which every left zero-divisor is also a right zero-divisor and conversely

2019 ◽  
Vol 18 (05) ◽  
pp. 1950096 ◽  
Author(s):  
E. Ghashghaei ◽  
M. Tamer Koşan ◽  
M. Namdari ◽  
T. Yildirim
Keyword(s):  
Zero Divisor ◽  
Matrix Ring ◽  
Triangular Matrix ◽  
Natural Generalization ◽  
Left Zero ◽  
Von Neumann ◽  
Von Neumann Regular ◽  
Triangular Matrix Ring ◽  
Upper Triangular Matrix ◽  
Upper Triangular Matrix Ring

A ring [Formula: see text] is called eversible if every left zero-divisor in [Formula: see text] is also a right zero-divisor and conversely. This class of rings is a natural generalization of reversible rings. It is shown that every eversible ring is directly finite, and a von Neumann regular ring is directly finite if and only if it is eversible. We give several examples of some important classes of rings (such as local, abelian) that are not eversible. We prove that [Formula: see text] is eversible if and only if its upper triangular matrix ring [Formula: see text] is eversible, and if [Formula: see text] is eversible then [Formula: see text] is eversible.

ZCn Rings and ZIn Rings

2012 ◽  
Vol 19 (04) ◽  
pp. 631-636
Author(s):  
Zhanping Wang ◽  
Limin Wang
Keyword(s):  
Matrix Ring ◽  
Triangular Matrix ◽  
Reduced Ring ◽  
Triangular Matrix Ring ◽  
Upper Triangular Matrix ◽  
Upper Triangular Matrix Ring

It is well known that the m × m upper triangular matrix ring over any ring is not ZIn (and so not ZCn) for m ≥ 2. In this paper, we find some ZCn subrings and ZIn subrings of the upper triangular matrix ring over a reduced ring.

Remarks on Centers of Rings

2021 ◽  
Vol 28 (01) ◽  
pp. 1-12
Author(s):  
Juan Huang ◽  
Hailan Jin ◽  
Tai Keun Kwak ◽  
Yang Lee ◽  
Zhelin Piao
Keyword(s):  
Matrix Ring ◽  
Triangular Matrix ◽  
Matrix Rings ◽  
Full Matrix ◽  
Triangular Matrix Ring ◽  
Upper Triangular Matrix ◽  
Upper Triangular Matrix Ring

It is proved that for matrices [Formula: see text], [Formula: see text] in the [Formula: see text] by [Formula: see text] upper triangular matrix ring [Formula: see text] over a domain [Formula: see text], if [Formula: see text] is nonzero and central in [Formula: see text] then [Formula: see text]. The [Formula: see text] by [Formula: see text] full matrix rings over right Noetherian domains are also shown to have this property. In this article we treat a ring property that is a generalization of this result, and a ring with such a property is said to be weakly reversible-over-center. The class of weakly reversible-over-center rings contains both full matrix rings over right Noetherian domains and upper triangular matrix rings over domains. The structure of various sorts of weakly reversible-over-center rings is studied in relation to the questions raised in the process naturally. We also consider the connection between the property of being weakly reversible-over-center and the related ring properties.

scholarly journals m-commuting Additive Maps on Upper Triangular Matrices Rings

2019 ◽  
pp. 505-514
Author(s):  
Driss Aiat Hadj Ahmed
Keyword(s):  
Commutative Ring ◽  
Matrix Ring ◽  
Triangular Matrix ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Additive Map ◽  
Triangular Matrix Ring ◽  
Upper Triangular Matrix ◽  
Additive Maps ◽  
Upper Triangular Matrix Ring

Let $T_{n}(R)$ be the upper triangular matrix ring over a unital commutative ring whose characteristic is not a divisor of $m$. Suppose that $f:T_{n}(R)\rightarrow T_{n}(R)$ is an additive map such that $X^{m}f(X)=f(X)X^{m}$ for all $x \in T_{n}(R),$ where $m\geq 1$ is an integer. We consider the problem of describing the form of the map $X \rightarrow f(X)$.

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