Let d be a non-zero derivation on a primitive ring R and ƒ(x1,…, xn) a homogeneous polynomial of degree m. We prove that the condition d(ƒ(r1,…, rn)t) = 0, for all r1,…, rn ∈ R, with t depending on r1,…, rn, forces R to be a finite dimensional central simple algebra and ƒ power-central valued on R. We also obtain bounds on [R : Z(R)] in terms of m.

Let R be a prime ring of characteristic not equal to two and let T be an automorphism of R. If U is a Lie ideal of R such that T is nontrivial on U and xxT — xTx is in the center of R for every x in U, then U is contained in the center of R.

In this paper we consider various degree two central polynomials with derivation, holding for Lie ideals in prime rings. The results give substantial generalizations of the existing ones on central and semi-centralizing derivations, and show essentially that there are no central identities of the form p(x,y) = c1xyD + C2XDy + c3yxD + C4yDx, where D is a nonzero derivation of the prime ring R.

Let R be a prime ring with invoution *, of characteristic 0, with skew elements K and extended centroid C. Let a ∈ K be such that (ad a)n =0 on K. It is shown that one of the following possibilities holds: (a) R is an order in a 4-dimensional central simple algebra, (b) there is a skew element λ in C such that , (c) * is of the first kind, n ≡ 0 or n ≡ 3 (mod 4), and . Examples are given illustrating (c).

A semiderivation of a ring R is an additive mapping f:R → R together with a function g:R → R such that f(xy) = f(x)g(y) + xf(y) = f(x)y + g(x)f(y) and f(g(x) ) = g(f(x)) for all x, y ∊ R. Motivating examples are derivations and mappings of the form x → x — g(x), g a ring endomorphism. A semiderivation f of R is centralizing on an ideal U if [f(u), u] is central for all u ∊ U. For R prime of char. ≠2, U a nonzero ideal of R, and 0 ≠ f a semiderivation of R we prove: (1) if f is centralizing on U then either R is commutative or f is essentially one of the motivating examples, (2) if [f(U), f(U) ] is central then R is commutative.

In this paper we prove the following result: let R be a prime ring with no non-zero nil left ideals whose characteristic is different from 2 and let U be a non central Lie ideal of R.

If d ≠ 0 is a derivation of R such that d(u) is invertible or nilpotent for all u ∈ U, then either R is a division ring or R is the 2 X 2 matrices over a division ring. Moreover in the last case if the division ring is non commutative, then d is an inner derivation of R.

Let R be a ring which possesses a unit element, a Lie ideal U ⊄ Z, and a derivation d such that d(U) ≠ 0 and d(u) is 0 or invertible, for all u ∈ U. We prove that R must be either a division ring D or D2, the 2 X 2 matrices over a division ring unless d is not inner, R is not semiprime, and either 2R or 3R is 0. We also examine for which division rings D, D2 can possess such a derivation and study when this derivation must be inner.

Let R be a prime ring with involution satisfying x/2 ∊ R whenever x ∊ R. Assume that R has two nontrivial symmetric idempotents e1, e2 whose sum is not 1, and that the subrings determined by e1, e2, 1 — (e1 + e2) are not orders in simple rings of dimension at most 4 over their centers. Then if L is a Lie derivation of the skew elements K into R there exists a subring A of R, A ⊆ , a derivation D :A → RC, the central closure of R, and a mapping T:R → C, satisfying L = D + T on K and = 0.

Let R be a ring with center Z, and S a nonempty subset of R. A mapping F from R to R is called centralizing on S if [x, F(x)] ∊ Z for all x ∊ S. We show that a semiprime ring R must have a nontrivial central ideal if it admits an appropriate endomorphism or derivation which is centralizing on some nontrivial one-sided ideal. Under similar hypotheses, we prove commutativity in prime rings.

Let R be a prime algebra over a field F and let d be a non-zero derivation in R such that for every x ∊ R, d(x) is algebraic over F of bounded degree. Then R is a primitive ring with a minimal right ideal eR, where e2 = e and eRe is a finite dimensional central division algebra.

A relation between Goldie dimensions of a modular lattice L and its sublattice LG of fixed points under a finite group G of automorphisms of L is obtained. The method used also gives a relation between ACC (DCC) for L and for LG. The results obtained are applied to rings and modules.

Let R be a prime ring and U be a nonzero ideal or quadratic Jordan ideal of R. If L is a nontrivial automorphism or derivation of R such thatuL(u)—L(u)u is in the center of R for every u in U, then the ring R is commutative.

Let R be a prime ring and d≠0 a derivation of R. We examine the relationship between the structure of R and that of d(R). We prove that if R is an algebra over a commutative ring A such that d(R) is a finitely generated submodule then R is an order in a simple algebra finite dimensional over its center.

It is shown that the nilpotency of a derivation on a 2-torsion free semiprime ring is always an odd number. Examples are provided to show the necessity of the assumptions.

In this paper we examine the nature of rings R with unit having an automorphism ϕ ≠ 1 such that x — ϕ(x) is 0 or invertible for every x ∊ R. We show that the only examples of such rings are R = D, R = D2, and R = D ⊕ D, where D is a division ring. Furthermore, for the case D2, we describe the division rings that are possible.