Results 121 to 130 of about 335,227 (157)
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Metaideals in Commutative Rings
Algebra Colloquium, 2005New examples of metaideals in commutative rings are constructed. It is proved that metaideals of a commutative ring form a sublattice of the lattice of all subrings, and for any subring A of a commutative ring P, there exists the largest subring Mid P (A) (called metaidealizer) in which A is a metaideal. Metaidealizers in several cases are described.
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On Commutative Splitting Rings
Proceedings of the London Mathematical Society, 1970Abstract not ...
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On Commutativity of Rings With Derivations
Results in Mathematics, 2002Let \(R\) be a ring, \(S\) a nonempty subset of \(R\), and \(Z\) the center of \(R\). For \(x,y\in R\) denote \(xy- yx\) by \([x, y]\) and \(xy + yx\) by \(x\circ y\). Let \(d\) be a derivation on \(R\). For prime \(R\) and \(S\) either an ideal or a Lie ideal, the authors study commutativity under the assumption that one of the following holds for all
Ashraf, Mohammad, Nadeem-ur-Rehman
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Commutativity of Rings with Constraints on Commutators, II
Results in Mathematics, 2000[For part I see ibid. 5, 123-131 (1985; Zbl 0606.16023).] The author proves commutativity of an associative ring \(R\) satisfying one of the following conditions: (1) for each \(x,y\in R\) there exists a co-monic polynomial \(p(t)\in tZ[t]\), such that \([x,y]=[x,y](p(xy)-p(yx))\); (2) for each \(x,y\in R\) there exist \(p(t),q(t)\in tZ[t]\) with \(q(t)
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Journal of Algebra and Its Applications, 2014
A ring is right tall if every non-noetherian right module contains a proper non-noetherian submodule. We prove a ring-theoretical criterion of tall commutative rings. Besides other examples which illustrate limits of proven necessary and sufficient conditions, we construct an example of a tall commutative ring that is non-max.
Penk, Tomáš, Žemlička, Jan
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A ring is right tall if every non-noetherian right module contains a proper non-noetherian submodule. We prove a ring-theoretical criterion of tall commutative rings. Besides other examples which illustrate limits of proven necessary and sufficient conditions, we construct an example of a tall commutative ring that is non-max.
Penk, Tomáš, Žemlička, Jan
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Commutativity of rings with powers commuting on subsets
Mathematical Journal of Okayama University, 1997Let \(R\) denote a ring with 1; let \(w=w(X,Y)\) denote a word, possibly 1, in two noncommuting indeterminates; and let \(n\) be a positive integer. The elements \(x,y\in R\) are said to satisfy condition \(a(w,n)\) (resp. \(b(w,n)\)) if \(w(x,y)[x^n,y^n]=0\) (resp. \(w(x,y)((xy)^n-(yx)^n)=0\)). Define \(A\subseteq R\) to be a \(P\)-subset if for each \
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Mathematics of the USSR-Sbornik, 1976
This paper deals with one-dimensional (commutative) rings without nilpotent elements such that every ideal is generated by three elements. It is shown that in such rings the square of every ideal is invertible, i.e. divides its multiplier ring. In addition, every ideal is distinguished, in the sense that on localization at any maximal ideal it becomes ...
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This paper deals with one-dimensional (commutative) rings without nilpotent elements such that every ideal is generated by three elements. It is shown that in such rings the square of every ideal is invertible, i.e. divides its multiplier ring. In addition, every ideal is distinguished, in the sense that on localization at any maximal ideal it becomes ...
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A Commutativity Condition for Rings
Canadian Journal of Mathematics, 1976The object of this paper is to prove the following theorem, a special case of which was previously explored in [1].THEOREM. Let R be any associative ring with the property that(†) for each x,y ∊ R, there exist integers m,n ≧ I for which xy = ymxn.
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On the commutativity of non-associative rings
Publicationes Mathematicae Debrecen, 2022\textit{E. C. Johnsen}, \textit{D. L. Outcalt}, and \textit{A. Yaqub} proved that a (not necessarily associative) ring with identity satisfying \((xy)^2 = x^2y^2\) for all \(x,y\) is commutative [Am. Math. Mon. 75, 288--289 (1968; Zbl 0162.33602)].
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On Commuting Rings Of Endomorphisms
Canadian Journal of Mathematics, 1956Various problems concerning the general theory of centralizers of modules which are not assumed to be completely reducible have been discussed by Fitting (3), Brauer (2), and Nakayama. In this paper we present a new approach to some of these questions, which has its origin in Weyl's discussion (15) of the centralizer of a finite group of collineations.
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