Generalized derivations and commutativity of rings with involution.
Let be a prime ring with its Utumi ring of quotients and extended centroid . Suppose that is a generalized derivation of and is a noncentral Lie ideal of such that for all , where is a fixed integer. Then one of the following holds:
We determine the Hochschild homology and cohomology of the generalized Weyl algebras of rank one which are of ‘quantum’ type in all but a few exceptional cases.
Let M be a 2 and 3-torsion free prime Γ-ring, d a nonzero derivation on M and U a nonzero Lie ideal of M. In this paper it is proved that U is a central Lie ideal of M if d satisfies one of the following (i) d(U)⊂ Z, (ii) d(U)⊂ U and d²(U)=0, (iii) d(U)⊂ U, d²(U)⊂ Z.
We study McCoy’s theorem to the skew Hurwitz series ring for some different classes of rings such as: semiprime rings, APP rings and skew Hurwitz serieswise quasi-Armendariz rings. Moreover, we establish an equivalence relationship between a right zip ring and its skew Hurwitz series ring in case when a ring satisfies McCoy’s theorem of skew Hurwitz series.
Let be a -torsion free -prime ring, a derivation which commutes with and a -Jordan ideal and a subring of . In this paper, it is shown that if either acts as a homomorphism or as an anti-homomorphism on , then or . Furthermore, an example is given to demonstrate that the -primeness hypothesis is not superfluous.
Let be a ring and an endomorphism of . We give a generalization of McCoy’s Theorem [ Annihilators in polynomial rings, Amer. Math. Monthly 64 (1957), 28–29] to the setting of skew polynomial rings of the form . As a consequence, we will show some results on semicommutative and -skew McCoy rings. Also, several relations among McCoyness, Nagata extensions and Armendariz rings and modules are studied.