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On Prime and Semiprime Rings with Derivations
Algebra Colloquium, 2006Let R be a ring and S a nonempty subset of R. A mapping f: R → R is called commuting on S if [f(x),x] = 0 for all x ∈ S. In this paper, firstly, we generalize the well-known result of Posner related to commuting derivations on prime rings. Secondly, we show that if R is a semiprime ring and I is a nonzero ideal of R, then a derivation d of R is ...
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On the Adjoint Group of Semiprime Rings
Communications in Algebra, 2006An associative ring R, not necessarily with a unity, is called semiprime if it has no nonzero nilpotent ideal. It is proved that in the adjoint group of a semiprime ring R every soluble-by-finite normal subgroup centralizes the Jacobson radical of R. In particular, if R is a semiprime ring with unity, then the same result holds for the multiplicative ...
CATINO, Francesco +2 more
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Semiprime Rings with Hypercentral Derivations
Canadian Mathematical Bulletin, 1995AbstractLetRbe a semiprime ring with a derivationd, λ a left ideal ofRandk, ntwo positive integers. Suppose that[d(xn),xn]k= 0 for allx∊ λ. Then [λ,R]d(R)= 0. That is, there exists a central idempotente∊U, the left Utumi quotient ring ofR, such thatdvanishes identically oneUand λ(l —e) is central in (1 —e ...
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On Jordan Structure in Semiprime Rings
Canadian Journal of Mathematics, 1976A remarkable theorem of Herstein [1, Theorem 2] of which we have made several uses states: If R is a semiprime ring of characteristic different from 2 and if U is both a Lie ideal and a subring of R then either U ⊂ Z (the centre of R) or U contains a nonzero ideal of R. In a recent paper [3] Herstein extends the above mentioned result significantly and
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THE SEMIPRIMENESS OF SEMIGROUP RINGS
JP Journal of Algebra, Number Theory and Applications, 2021Hirano, Yasuyuki +2 more
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Mathematical Journal of Okayama University, 1995
In an earlier paper, the author developed a theory that in a semiprime torsion free ring, there is an essential direct sum of three completely unique and algebraically very different types of ideals, one of which is discrete and the others are continuous.
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In an earlier paper, the author developed a theory that in a semiprime torsion free ring, there is an essential direct sum of three completely unique and algebraically very different types of ideals, one of which is discrete and the others are continuous.
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On Skew Derivations in Semiprime Rings
Algebras and Representation Theory, 2012Let \(R\) be a ring with center \(Z(R)\), and let \(\sigma\) be an endomorphism of \(R\). An additive map \(\delta\colon R\to R\) is called a \(\sigma\)-derivation if \(\delta(xy)=\sigma(x)\delta(y)+\delta(x)y\) for all \(x,y\in R\). The principal result of the paper, which generalizes a result of the reviewer and \textit{M. N. Daif} [Can. Math.
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On derivation of semiprime rings
2012The paper purports to prove several commutativity theorems for prime or semiprime rings satisfying certain constraints involving derivations, one such being that for some derivation \(d\), \(xyx+d(xyx)=x^2y+d(x^2y)\) for all \(x,y\in R\). Unfortunately the proofs are wrong.
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Functional equations related to higher derivations in semiprime rings
Open Mathematics, 2021O H Ezzat
exaly
1973
A ring S is a (classical) right quotient ring of a subring T if every regular element a ∈ T has an inverse in S and $$ S = \{ a{b^{ - 1}}|a,b \in T,b\;{\text{reular}}\} $$ Then T is an order in S (cf. 7.21). The following condition is necessary and sufficient for a ring T to possess a classical quotient ring: If a, b ∈ T, and if b is regular ...
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A ring S is a (classical) right quotient ring of a subring T if every regular element a ∈ T has an inverse in S and $$ S = \{ a{b^{ - 1}}|a,b \in T,b\;{\text{reular}}\} $$ Then T is an order in S (cf. 7.21). The following condition is necessary and sufficient for a ring T to possess a classical quotient ring: If a, b ∈ T, and if b is regular ...
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