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EXTENSIONS OF STRONGLY π-REGULAR RINGS
Huanyin Chen,Handan Kose,Yosum Kurtulmaz 대한수학회 2014 대한수학회보 Vol.51 No.2
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].
EXTENSIONS OF STRONGLY π-REGULAR RINGS
Chen, Huanyin,Kose, Handan,Kurtulmaz, Yosum Korean Mathematical Society 2014 대한수학회보 Vol.51 No.2
An ideal I of a ring R is strongly ${\pi}$-regular if for any $x{\in}I$ there exist $n{\in}\mathbb{N}$ and $y{\in}I$ such that $x^n=x^{n+1}y$. We prove that every strongly ${\pi}$-regular ideal of a ring is a B-ideal. An ideal I is periodic provided that for any $x{\in}I$ there exist two distinct m, $n{\in}\mathbb{N}$ such that $x^m=x^n$. Furthermore, we prove that an ideal I of a ring R is periodic if and only if I is strongly ${\pi}$-regular and for any $u{\in}U(I)$, $u^{-1}{\in}\mathbb{Z}[u]$.
SYMMETRICITY AND REVERSIBILITY FROM THE PERSPECTIVE OF NILPOTENTS
Harmanci, Abdullah,Kose, Handan,Ungor, Burcu Korean Mathematical Society 2021 대한수학회논문집 Vol.36 No.2
In this paper, we deal with the question that what kind of properties does a ring gain when it satisfies symmetricity or reversibility by the way of nilpotent elements? By the motivation of this question, we approach to symmetric and reversible property of rings via nilpotents. For symmetricity, we call a ring R middle right-(resp. left-)nil symmetric (mr-nil (resp. ml-nil) symmetric, for short) if abc = 0 implies acb = 0 (resp. bac = 0) for a, c ∈ R and b ∈ nil(R) where nil(R) is the set of all nilpotent elements of R. It is proved that mr-nil symmetric rings are abelian and so directly finite. We show that the class of mr-nil symmetric rings strictly lies between the classes of symmetric rings and weak right nil-symmetric rings. For reversibility, we introduce left (resp. right) N-reversible ideal I of a ring R if for any a ∈ nil(R), b ∈ R, being ab ∈ I implies ba ∈ I (resp. b ∈ nil(R), a ∈ R, being ab ∈ I implies ba ∈ I). A ring R is called left (resp. right) N-reversible if the zero ideal is left (resp. right) N-reversible. Left N-reversibility is a generalization of mr-nil symmetricity. We exactly determine the place of the class of left N-reversible rings which is placed between the classes of reversible rings and CNZ rings. We also obtain that every left N-reversible ring is nil-Armendariz. It is observed that the polynomial ring over a left N-reversible Armendariz ring is also left N-reversible.
Strongly Clean Matrices Over Power Series
Chen, Huanyin,Kose, Handan,Kurtulmaz, Yosum Department of Mathematics 2016 Kyungpook mathematical journal Vol.56 No.2
An $n{\times}n$ 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 $A(x){\in}M_n(R[[x]])$. We prove, in this note, that $A(x){\in}M_n(R[[x]])$ is strongly clean if and only if $A(0){\in}M_n(R)$ is strongly clean. Strongly clean matrices over quotient rings of power series are also determined.
Certain Clean Decompositions for Matrices over Local Rings
Yosum Kurtulmaz,Huanyin Chen,Handan Kose 경북대학교 자연과학대학 수학과 2023 Kyungpook mathematical journal Vol.63 No.4
An element a ∈ R is strongly rad-clean provided that there exists an idem potent e ∈ R such that a-e ∈ U(R), ae = ea and eae ∈ J(eRe). In this article, we completely determine when a 2x2 matrix over a commutative local ring is strongly rad clean. An application to matrices over power-series is also given.