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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.
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 $u{\in}U(I)$, $u^{-1}{\in}\mathbb{Z}[u]$.
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.
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].