Results 11 to 20 of about 46 (46)
Structure of weakly periodic rings with potent extended commutators
International Journal of Mathematics and Mathematical Sciences, Volume 25, Issue 5, Page 299-304, 2001., 2001 A well‐known theorem of Jacobson (1964, page 217) asserts that a ring R with the property that, for each x in R, there exists an integer n(x) > 1 such that xn(x) = x is necessarily commutative. This theorem is generalized to the case of a weakly periodic ring R with a “sufficient” number of potent extended commutators.
Adil Yaqub
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Generalized periodic and generalized Boolean rings
International Journal of Mathematics and Mathematical Sciences, Volume 26, Issue 8, Page 457-465, 2001., 2001 We prove that a generalized periodic, as well as a generalized Boolean, ring is either commutative or periodic. We also prove that a generalized Boolean ring with central idempotents must be nil or commutative. We further consider conditions which imply the commutativity of a generalized periodic, or a generalized Boolean, ring.
Howard E. Bell, Adil Yaqub
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International Journal of Mathematics and Mathematical Sciences, Volume 24, Issue 1, Page 55-57, 2000., 2000 For prime rings R, we characterize the set U∩CR([U, U]), where U is a right ideal of R; and we apply our result to obtain a commutativity‐or‐finiteness theorem. We include extensions to semiprime rings.
Howard E. Bell
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On structure of certain periodic rings and near‐rings
International Journal of Mathematics and Mathematical Sciences, Volume 24, Issue 10, Page 667-672, 2000., 2000 The aim of this work is to study a decomposition theorem for rings satisfying either of the properties xy = xpf(xyx)xq or xy = xpf(yxy)xq, where p = p(x, y), q = q(x, y) are nonnegative integers and f(t) ∈ tℤ[t] vary with the pair of elements x, y, and further investigate the commutativity of such rings.
Moharram A. Khan
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Commutativity results for semiprime rings with derivations
International Journal of Mathematics and Mathematical Sciences, Volume 21, Issue 3, Page 471-474, 1998., 1996 We extend a result of Herstein concerning a derivation d on a prime ring R satisfying [d(x), d(y)] = 0 for all x, y ∈ R, to the case of semiprime rings. An extension of this result is proved for a two‐sided ideal but is shown to be not true for a one‐sided ideal.
Mohammad Nagy Daif
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Commutativity of one sided s‐unital rings through a Streb′s result
International Journal of Mathematics and Mathematical Sciences, Volume 20, Issue 2, Page 267-270, 1997., 1995 The main theorem proved in the present paper states as follows “Let m, k, n and s be fixed non‐negative integers such that k and n are not simultaneously equal to 1 and R be a left (resp right) s‐unital ring satisfying [(xmyk)n−xsy,x]=0 (resp [(xmyk)n−yxs,x]=0) Then R is commutative.” Further commutativity of left s‐unital rings satisfying the ...
Murtaza A. Quadri +2 more
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International Journal of Mathematics and Mathematical Sciences, Volume 19, Issue 1, Page 87-92, 1996., 1995 Let R be a ring, and let N and C denote the set of nilpotents and the center of R, respectively. R is called generalized periodic if for every x ∈ R\(N ⋃ C), there exist distinct positive integers m, n of opposite parity such that xn − xm ∈ N ⋂ C. We prove that a generalized periodic ring always has the set N of nilpotents forming an ideal in R.
Howard E. Bell, Adil Yaqub
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Direct sums of J‐rings and radical rings
International Journal of Mathematics and Mathematical Sciences, Volume 18, Issue 3, Page 531-534, 1995., 1994 Let R be a ring, J(R) the Jacobson radical of R and P the set of potent elements of R. We prove that if R satisfies (∗) given x, y in R there exist integers m = m(x, y) > 1 and n = n(x, y) > 1 such that xmy = xyn and if each x ∈ R is the sum of a potent element and a nilpotent element, then N and P are ideals and R = N ⊕ P.
Xiuzhan Guo
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International Journal of Mathematics and Mathematical Sciences, Volume 17, Issue 4, Page 667-670, 1994., 1993 Let R be a ring, and let C denote the center of R. R is said to have a prime center if whenever ab belongs to C then a belongs to C or b belongs to C. The structure of certain classes of these rings is studied, along with the relation of the notion of prime centers to commutativity. An example of a non‐commutative ring with a prime center is given.
Hazar Abu-Khuzam, Adil Yaqub
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Lie ideals and derivations of σ-prime rings”, [PDF]
, 2007 Let R be a 2-torsion free σ-prime ring with involution σ, U a nonzero Lie ideal of R and d : R −→ R a nonzero derivation commuting with σ.
L Oukhtite, S Salhi
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