Browsing by Author "Kose, H."

Now showing 1 - 7 of 7

Open Access
Extensions of strongly π-regular rings
(Korean Mathematical Society, 2014) Chen, H.; Kose, H.; Kurtulmaz, Y.
An ideal I of a ring R is strongly π -regular if for any x ∈ I there exist n ∈ ℕ and y ∈ I such that xn = xn+1y. We prove that every strongly π -regular ideal of a ring is a B-ideal. An ideal I is periodic provided that for any x ∈ I there exist two distinct m, n ∈ N such that xm = xn. Furthermore, we prove that an ideal I of a ring R is periodic if and only if I is strongly π -regular and for any u ∈ U(I), u-1 ∈ ℤ[u]. © 2014 Korean Mathematical Society.
Open Access
Factorizations of matrices over projective-free Rings
(World Scientific Publishing Co. Pte Ltd, 2016) Chen, H.; Kose, H.; Kurtulmaz, Y.
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.
Open Access
On feckly clean rings
(World Scientific Publishing, 2015) Chen, H.; Kose, H.; Kurtulmaz, Y.
A ring R is feckly clean provided that for any a R there exists an element e R and a full element u R such that a = e + u, eR(1 - e) J(R). We prove that a ring R is feckly clean if and only if for any a R, there exists an element e R such that V(a) V(e), V(1 - a) V(1 - e) and eR(1 - e) J(R), if and only if for any distinct maximal ideals M and N, there exists an element e R such that e M, 1 - e N and eR(1 - e) J(R), if and only if J-spec(R) is strongly zero-dimensional, if and only if Max(R) is strongly zero-dimensional and every prime ideal containing J(R) is contained in a unique maximal ideal. More explicit characterizations are also discussed for commutative feckly clean rings. © 2015 World Scientific Publishing Company.
Open Access
On π-Morphic modules
(Hacettepe University, Department of Mathematics, 2013) Harmanci, A.; Kose, H.; Kurtulmaz, Yosum
Let R be an arbitrary ring with identity and M be a right R-module with S = End(MR). Let f ∈ S. f is called π-morphic if M/f n(M) ∼=rM(fn) for some positive integer n. A module M is called π-morphic if every f ∈ S is π-morphic. It is proved that M is π-morphic and image-projective if and only if S is right π-morphic and M generates its kernel. S is unit-π-regular if and only if M is π-morphic and π-Rickart if and only if M is π-morphic and dual π-Rickart. M is π-morphic and image-injective if and only if S is left π-morphic and M cogenerates itscokernel.
Open Access
Strongly clean matrices over power series
(Kyungpook National University, 2016) Chen, H.; Kose, H.; Kurtulmaz, Y.
An n × 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.
Open Access
Strongly clean triangular matrix rings with endomorphisms
(Springer, 2015) Chen, H.; Kose, H.; Kurtulmaz, Y.
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.
Open Access
Sytongly P-clean Rings and Matrices
(Elsevier, 2014) Chen, H.; Kose, H.; Kurtulmaz, Y.
Abstract. 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 sufficient conditions under which a ring R is strongly P-clean. Many characterizations of such rings are obtained. The criteria on strong P-cleanness of 2 × 2 matrices over commutative projective-free rings are also determined.