13 research outputs found

### Factorizations of Matrices Over Projective-free Rings

An element of a ring $R$ is called strongly $J^{\#}$-clean provided that it
can be written as the sum of an idempotent and an element in $J^{\#}(R)$ that
commute. We characterize, in this article, the strongly $J^{\#}$-cleanness of
matrices over projective-free rings. These extend many known results on
strongly clean matrices over commutative local rings

### Strongly clean matrices over power series

An n ï¿½ n matrix A over a commutative ring is strongly clean provided that it can be written as the sum of an idempotent matrix and an invertible matrix that commute. Let R be an arbitrary commutative ring, and let A(x) ∈ Mn (R[[x]]). We prove, in this note, that A(x) ∈ Mn (R[[x]]) is strongly clean if and only if A(0) ∈ Mn(R) is strongly clean. Strongly clean matrices over quotient rings of power series are also determined. ï¿½ Kyungpook Mathematical Journal

### Strongly clean triangular matrix rings with endomorphisms

A ring R is strongly clean provided that every element in R is the sum of an idempotent and a unit that commutate. Let Tn(R; σ) be the skew triangular matrix ring over a local ring R where σ is an endomorphism of R. We show that T2(R; σ) is strongly clean if and only if for any aϵ 1+J(R); b ϵ J(R), la -rσ (b): R→ R is surjective. Further, T3(R; σ) is strongly clean if la-rσ (b); la-rσ2 (b) and lb-rσ (a)are surjective for any a ϵ U(R); b ϵ J(R). The necessary condition for T3(R; σ) to be strongly clean is also obtained. © 2015 Iranian Mathematical Society

### Symmetric modules over their endomorphism rings

Let R be an arbitrary ring with identity and M a right
R-module with S=EndR(M). In this paper, we study right
R-modules M having the property for f,g∈EndR(M) and
for m∈M, the condition fgm=0 implies gfm=0. We prove
that some results of symmetric rings can be extended to symmetric
modules for this general setting

### Symmetric modules over their endomorphism rings

Let R be an arbitrary ring with identity and M a right R-module with S = EndR(M). In this paper, we study right R-modules M having the property for f, g ∈ EndR(M) and for m ∈ M, the condition fgm = 0 implies gfm = 0. We prove that some results of symmetric rings can be extended to symmetric modules for this general setting. © Journal “Algebra and Discrete Mathematics”

### Very cleanness of generalized matrices

An element a in a ring R is very clean in case there exists an idempotent e ∈ R such that ae = ea and either a – e or a + e is invertible. An element a in a ring R is very J-clean provided that there exists an idempotent e ∈ R such that ae = ea and either a – e ∈ J(R) or a + e ∈ J(R). Let R be a local ring, and let s ∈ C(R). We prove that A ∈ Ks(R) is very clean if and only if A ∈ U(Ks(R)), I ± A ∈ U(Ks(R)) or A ∈ Ks(R) is very J-clean. © 2017 Iranian Mathematical Society

### Sytongly P-clean Rings and Matrices

Cataloged from PDF version of article.An element of a ring
R
is strongly
P
-clean provided that it can
be written as the sum of an idempotent and a strongly nilpotent element that
commute. A ring
R
is strongly P-clean in case each of its elements is strongly P-clean. We investigate, in this article, the necessary and su cient conditions
under which a ring R is strongly P-clean. Many characterizations of such rings are obtained. The criteria on strong
P-cleanness of 2x2 matrices over
commutative projective-free rings are also determined

### Factorizations of Matrices over Projective-free Rings

An element of a ring R is called strongly J#-clean provided that it can be written as the sum of an idempotent and an element in J#(R) that commute. In this paper, we characterize the strong J#-cleanness of matrices over projective-free rings. This extends many known results on strongly clean matrices over commutative local rings. © 2016 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and Suzhou University

### Factorizations of Matrices over Projective-free Rings

WOS: 000367819000003An element of a ring R is called strongly J(#)-clean provided that it can be written as the sum of an idempotent and an element in J(#)(R) that commute. In this paper, we characterize the strong J(#)-cleanness of matrices over projective-free rings. This extends many known results on strongly clean matrices over commutative local rings.Scientific and Technological Research Council of Turkey (2221 Visiting Scientists Fellowship Programme)Turkiye Bilimsel ve Teknolojik Arastirma Kurumu (TUBITAK); Natural Science Foundation of Zhejiang Province, ChinaNatural Science Foundation of Zhejiang Province [LY13A010019]This research was supported by the Scientific and Technological Research Council of Turkey (2221 Visiting Scientists Fellowship Programme) and the Natural Science Foundation of Zhejiang Province (LY13A010019), China