نتایج جستجو برای: strongly $J_n$-clean ring, $2imes 2$ matrix
تعداد نتایج: 3057630 فیلتر نتایج به سال:
a ring $r$ is a strongly clean ring if every element in $r$ is the sum of an idempotent and a unit that commutate. we construct some classes of strongly clean rings which have stable range one. it is shown that such cleanness of $2 imes 2$ matrices over commutative local rings is completely determined in terms of solvability of quadratic equations.
A ring $R$ is a strongly clean ring if every element in $R$ is the sum of an idempotent and a unit that commutate. We construct some classes of strongly clean rings which have stable range one. It is shown that such cleanness of $2 imes 2$ matrices over commutative local rings is completely determined in terms of solvability of quadratic equations.
A ring $R$ is strongly clean provided that every element in $R$ is the sum of an idempotent and a unit that commutate. Let $T_n(R,sigma)$ be the skew triangular matrix ring over a local ring $R$ where $sigma$ is an endomorphism of $R$. We show that $T_2(R,sigma)$ is strongly clean if and only if for any $ain 1+J(R), bin J(R)$, $l_a-r_{sigma(b)}: Rto R$ is surjective. Furt...
a ring $r$ is strongly clean provided that every element in $r$ is the sum of an idempotent and a unit that commutate. let $t_n(r,sigma)$ be the skew triangular matrix ring over a local ring $r$ where $sigma$ is an endomorphism of $r$. we show that $t_2(r,sigma)$ is strongly clean if and only if for any $ain 1+j(r), bin j(r)$, $l_a-r_{sigma(b)}: rto r$ is surjective. furt...
let a; b; k 2 k and u ; v 2 u(k). we show for any idempotent e 2 k, ( a 0 b 0 ) is e-clean i ( a 0 u(vb + ka) 0 ) is e-clean and if ( a 0 b 0 ) is 0-clean, ( ua 0 u(vb + ka) 0 ) is too.
The classes of clean and nil-clean rings are closed with respect standard constructions as direct products and (triangular) matrix rings, cf. [12] resp. [4], while the classes of weakly (nil-)clean rings are not closed under these constructions. Moreover, while all matrix rings over fields are clean, [12] when we consider nil-clean rings there are strongly restrictions: if a matrix ring over a ...
A ring R is called strongly clean if every element of R is the sum of a unit and an idempotent that commute with each other. A recent result of Borooah, Diesl and Dorsey [3] completely characterized the commutative local rings R for which Mn(R) is strongly clean. For a general local ring R and n > 1, however, it is unknown when the matrix ring Mn(R) is strongly clean. Here we completely determi...
Let R be a commutative local ring. It is proved that R is Henselian if and only if each R-algebra which is a direct limit of module finite R-algebras is strongly clean. So, the matrix ring Mn(R) is strongly clean for each integer n > 0 if R is Henselian and we show that the converse holds if either the residue class field of R is algebraically closed or R is an integrally closed domain or R is ...
We show that if $R$ is a ring with an arbitrary idempotent $e$ such that $eRe$ and $(1-e)R(1-e)$ are both strongly nil-clean rings, then $R/J(R)$ is nil-clean. In particular, under certain additional circumstances, $R$ is also nil-clean. These results somewhat improves on achievements due to Diesl in J. Algebra (2013) and to Koc{s}an-Wang-Zhou in J. Pure Appl. Algebra (2016). ...
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