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.

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.

A ring R is called an EC-ring if for each x, y ∊ R, there exist distinct positive integers m, n such that the extended commutators [x, y]m and [x, y]n are equal. We show that in certain EC-rings, the commutator ideal is periodic; we establish commutativity of arbitrary EC-domains; we prove that a ring R is commutative if for each x, y ∊ R, there exists n > 1 for which [x, y] = [x, y]n.

Let R be a periodic ring, N the set of nilpotents, and D the set of right zero divisors of R. Suppose that (i) N is commutative, and (ii) every x in R can be uniquely written in the form x = e + a, where e2 = e and a ∊ N. Then N is an ideal in R and R/N is a Boolean ring. If (i) is satisfied but (ii) is now assumed to hold merely for those elements x ∊ D, and if 1 ∊ R, then N is still an ideal in R and R/N is a subdirect sum of fields. It is further shown that if (i) is satisfied but (ii) is replaced by: "every right zero divisor is either nilpotent or idempotent," and if 1 ∊ R, then N is still an ideal in R and R/N is either a Boolean ring or a field.

If O → A → C → B → O is a short exact sequence of finitely generated modules over a Noetherian Pi-algebra then we show that GK(C) = max{GK(A), GK(B)}.

The following theorem is proved: Suppose R is a ring with identity which satisfies the identities xkyk = ykxk and xlyl = ylxl, where k and l are positive relatively prime integers. Then R is commutative. This theorem also holds for a group G. Furthermore, examples are given which show that neither R nor G need be commutative if either of the above identities is dropped. The proof of the commutativity of R uses the fact that G is commutative, where G is taken to be the group R* of units in R.