A ring R with identity is called strongly clean if every element of R is the sum of an idempotent... more A ring R with identity is called strongly clean if every element of R is the sum of an idempotent and a unit that commute. For a commutative local ring R, n = 3, 4, and m, k, s ∈ ގ it is proved that ލ n (R) is strongly clean if and only if ލ n (R[[x]]) is strongly clean if and only if ލ n (R[[x 1 , x 2 ,. .. , x m ]]) is strongly clean if and only if ލ n (R[x] (x k)) is strongly clean if and only if ލ n (R[
A ring R with identity is called “clean” if for every element a ∈ R there exist an idempotent e a... more A ring R with identity is called “clean” if for every element a ∈ R there exist an idempotent e and a unit u in R such that a = e + u. Let C(R) denote the center of a ring R and g(x) be a polynomial in the polynomial ring C(R)[x]. An element r ∈ R is called “g(x)-clean” if r = s + u where g(s) = 0 and u is a unit of R and R is g(x)-clean if every element is g(x)-clean. Clean rings are g(x)-clean where g(x) ∈ (x − a)(x − b)C(R)[x] with a, b ∈ C(R) and b − a ∈ U(R); equivalent conditions for (x2 − 2x)-clean rings are obtained; and some properties of g(x)-clean rings are given.
A ring R is called strongly clean if every element of R is the sum of a unit and an idempotent th... more A ring R is called strongly clean if every element of R is the sum of a unit and an idempotent that commute. By SRC factorization, Borooah, Diesl, and Dorsey [3] completely determined when Mn(R) over a commutative local ring R is strongly clean. We generalize the notion of SRC factorization to commutative rings, prove that commutative n-SRC rings (n ≥ 2) are precisely the commutative local rings over which Mn(R) is strongly clean, and characterize strong cleanness of matrices over commutative projective-free rings having ULP. The strongly π-regular property (hence, strongly clean property) of Mn(C(X, C)) with X a P-space relative to C is also obtained where C(X, C) is the ring of complex valued continuous functions.
Let R be an associative ring with identity, C(R) denote the center of R, and g(x) be a polynomial... more Let R be an associative ring with identity, C(R) denote the center of R, and g(x) be a polynomial in the polynomial ring C(R)[x]. R is called strongly g(x)-clean if every element r ∈ R can be written as r = s + u with g(s) = 0, u a unit of R, and su = us. The relation between strongly g(x)-clean rings and strongly clean rings is determined, some general properties of strongly g(x)-clean rings are given, and strongly g(x)-clean rings generated by units are discussed.
Let R be an associative ring with identity. An element a ∈ R is called clean if a = e + u with e ... more Let R be an associative ring with identity. An element a ∈ R is called clean if a = e + u with e an idempotent and u a unit of R, and a is called strongly clean if, in addition, eu = ue. A ring R is clean if every element of R is clean, and R is strongly clean if every element of R is strongly clean. When is a matrix ring over a strongly clean ring strongly clean? Does a strongly clean ring have stable range one? For these open questions, we prove that 𝕄 n (C(X)) is strongly π-regular (hence, strongly clean) where C(X) is the ring of all real valued continuous functions on X with X a P-space; C(X) is clean iff it has stable range one; and a unital C*-algebra in which every unit element is self-adjoint is clean iff it has stable range one. The criteria for the ring of complex valued continuous functions C(X,ℂ) to be strongly clean is given.
Let R be an associative ring with identity. An element a ∈ R is called strongly clean if a = e + ... more Let R be an associative ring with identity. An element a ∈ R is called strongly clean if a = e + u with e 2 = e ∈ R, u a unit of R, and eu = ue. A ring R is called strongly clean if every element of R is strongly clean. Strongly clean rings were introduced by Nicholson [7]. It is unknown yet when a matrix ring over a strongly clean ring is strongly clean. Several articles discussed this topic when R is local or strongly-regular. In this note, necessary conditions for the matrix ring n R n > 1 over an arbitrary ring R to be strongly clean are given, and the strongly clean property of 2 RC 2 over the group ring RC 2 with R local is obtained.
A ring R with identity is called “clean” if for every element a ∈ R there exist an idempotent e a... more A ring R with identity is called “clean” if for every element a ∈ R there exist an idempotent e and a unit u in R such that a = e + u. Let C(R) denote the center of a ring R and g(x) be a polynomial in the polynomial ring C(R)[x]. An element r ∈ R is called “g(x)-clean” if r = s + u where g(s) = 0 and u is a unit of R and R is g(x)-clean if every element is g(x)-clean. Clean rings are g(x)-clean where g(x) ∈ (x − a)(x − b)C(R)[x] with a, b ∈ C(R) and b − a ∈ U(R); equivalent conditions for (x2 − 2x)-clean rings are obtained; and some properties of g(x)-clean rings are given.
Let R be an associative ring with identity, C(R) denote the center of R, and g(x) be a polynomial... more Let R be an associative ring with identity, C(R) denote the center of R, and g(x) be a polynomial in the polynomial ring C(R)[x]. R is called strongly g(x)-clean if every element r ∈ R can be written as r = s + u with g(s) = 0, u a unit of R, and su = us. The relation between strongly g(x)-clean rings and strongly clean rings is determined, some general properties of strongly g(x)-clean rings are given, and strongly g(x)-clean rings generated by units are discussed.
Let $R$ be an associative ring with identity, $C(R)$ denote the center of $R$, and $g(x)$ be a po... more Let $R$ be an associative ring with identity, $C(R)$ denote the center of $R$, and $g(x)$ be a polynomial in the polynomial ring $C(R)[x]$. $R$ is called strongly $g(x)$-clean if every element $r \in R$ can be written as $r=s+u$ with $g(s)=0$, $u$ a unit of $R$, and $su=us$. The relation between strongly $g(x)$-clean rings and strongly clean rings is determined, some general properties of strongly $g(x)$-clean rings are given, and strongly $g(x)$-clean rings generated by units are discussed.
A ring R with identity is called strongly clean if every element of R is the sum of an idempotent... more A ring R with identity is called strongly clean if every element of R is the sum of an idempotent and a unit that commute. For a commutative local ring R, n = 3, 4, and m, k, s ∈ ގ it is proved that ލ n (R) is strongly clean if and only if ލ n (R[[x]]) is strongly clean if and only if ލ n (R[[x 1 , x 2 ,. .. , x m ]]) is strongly clean if and only if ލ n (R[x] (x k)) is strongly clean if and only if ލ n (R[
A ring R with identity is called “clean” if for every element a ∈ R there exist an idempotent e a... more A ring R with identity is called “clean” if for every element a ∈ R there exist an idempotent e and a unit u in R such that a = e + u. Let C(R) denote the center of a ring R and g(x) be a polynomial in the polynomial ring C(R)[x]. An element r ∈ R is called “g(x)-clean” if r = s + u where g(s) = 0 and u is a unit of R and R is g(x)-clean if every element is g(x)-clean. Clean rings are g(x)-clean where g(x) ∈ (x − a)(x − b)C(R)[x] with a, b ∈ C(R) and b − a ∈ U(R); equivalent conditions for (x2 − 2x)-clean rings are obtained; and some properties of g(x)-clean rings are given.
A ring R is called strongly clean if every element of R is the sum of a unit and an idempotent th... more A ring R is called strongly clean if every element of R is the sum of a unit and an idempotent that commute. By SRC factorization, Borooah, Diesl, and Dorsey [3] completely determined when Mn(R) over a commutative local ring R is strongly clean. We generalize the notion of SRC factorization to commutative rings, prove that commutative n-SRC rings (n ≥ 2) are precisely the commutative local rings over which Mn(R) is strongly clean, and characterize strong cleanness of matrices over commutative projective-free rings having ULP. The strongly π-regular property (hence, strongly clean property) of Mn(C(X, C)) with X a P-space relative to C is also obtained where C(X, C) is the ring of complex valued continuous functions.
Let R be an associative ring with identity, C(R) denote the center of R, and g(x) be a polynomial... more Let R be an associative ring with identity, C(R) denote the center of R, and g(x) be a polynomial in the polynomial ring C(R)[x]. R is called strongly g(x)-clean if every element r ∈ R can be written as r = s + u with g(s) = 0, u a unit of R, and su = us. The relation between strongly g(x)-clean rings and strongly clean rings is determined, some general properties of strongly g(x)-clean rings are given, and strongly g(x)-clean rings generated by units are discussed.
Let R be an associative ring with identity. An element a ∈ R is called clean if a = e + u with e ... more Let R be an associative ring with identity. An element a ∈ R is called clean if a = e + u with e an idempotent and u a unit of R, and a is called strongly clean if, in addition, eu = ue. A ring R is clean if every element of R is clean, and R is strongly clean if every element of R is strongly clean. When is a matrix ring over a strongly clean ring strongly clean? Does a strongly clean ring have stable range one? For these open questions, we prove that 𝕄 n (C(X)) is strongly π-regular (hence, strongly clean) where C(X) is the ring of all real valued continuous functions on X with X a P-space; C(X) is clean iff it has stable range one; and a unital C*-algebra in which every unit element is self-adjoint is clean iff it has stable range one. The criteria for the ring of complex valued continuous functions C(X,ℂ) to be strongly clean is given.
Let R be an associative ring with identity. An element a ∈ R is called strongly clean if a = e + ... more Let R be an associative ring with identity. An element a ∈ R is called strongly clean if a = e + u with e 2 = e ∈ R, u a unit of R, and eu = ue. A ring R is called strongly clean if every element of R is strongly clean. Strongly clean rings were introduced by Nicholson [7]. It is unknown yet when a matrix ring over a strongly clean ring is strongly clean. Several articles discussed this topic when R is local or strongly-regular. In this note, necessary conditions for the matrix ring n R n > 1 over an arbitrary ring R to be strongly clean are given, and the strongly clean property of 2 RC 2 over the group ring RC 2 with R local is obtained.
A ring R with identity is called “clean” if for every element a ∈ R there exist an idempotent e a... more A ring R with identity is called “clean” if for every element a ∈ R there exist an idempotent e and a unit u in R such that a = e + u. Let C(R) denote the center of a ring R and g(x) be a polynomial in the polynomial ring C(R)[x]. An element r ∈ R is called “g(x)-clean” if r = s + u where g(s) = 0 and u is a unit of R and R is g(x)-clean if every element is g(x)-clean. Clean rings are g(x)-clean where g(x) ∈ (x − a)(x − b)C(R)[x] with a, b ∈ C(R) and b − a ∈ U(R); equivalent conditions for (x2 − 2x)-clean rings are obtained; and some properties of g(x)-clean rings are given.
Let R be an associative ring with identity, C(R) denote the center of R, and g(x) be a polynomial... more Let R be an associative ring with identity, C(R) denote the center of R, and g(x) be a polynomial in the polynomial ring C(R)[x]. R is called strongly g(x)-clean if every element r ∈ R can be written as r = s + u with g(s) = 0, u a unit of R, and su = us. The relation between strongly g(x)-clean rings and strongly clean rings is determined, some general properties of strongly g(x)-clean rings are given, and strongly g(x)-clean rings generated by units are discussed.
Let $R$ be an associative ring with identity, $C(R)$ denote the center of $R$, and $g(x)$ be a po... more Let $R$ be an associative ring with identity, $C(R)$ denote the center of $R$, and $g(x)$ be a polynomial in the polynomial ring $C(R)[x]$. $R$ is called strongly $g(x)$-clean if every element $r \in R$ can be written as $r=s+u$ with $g(s)=0$, $u$ a unit of $R$, and $su=us$. The relation between strongly $g(x)$-clean rings and strongly clean rings is determined, some general properties of strongly $g(x)$-clean rings are given, and strongly $g(x)$-clean rings generated by units are discussed.
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