Let R be a prime ring of characteristic not 2. Automorphisms α and β of R satisfying α ≠ β, α ≠ β−1, and α + α−1 = β + β-1 are characterized. This result is an algebraic analogue of some results for operator algebras.

Band sums of associative rings were introduced by Weissglass in 1973. The main theorem claims that the support of every Artinian band sum of rings is finite. This result is analogous to the well-known theorem on Artinian semigroup rings.

In this paper we prove algebraic generalizations of some results of C. J. K. Batty and A. B. Thaheem, concerned with the identity α + α−1 = β + β−1 where α and β are automorphisms of a C*-algebra. The main result asserts that if automorphisms α, β of a semiprime ring R satisfy α + α-1 = β + β−1 then there exist invariant ideals U1, U2 and U3 of R such that Ui ∩ Uj = 0, i ≠ j, U1 ⊕ U2 ⊕ U3 is an essential ideal, α = β on U1, α = β−1 on U2, and α2 = β2 = α−2 on U3. Furthermore, if the annihilator of any ideal in R is a direct summand (in particular, if R is a von Neumann algebra), then U1 ⊕ U2 ⊕ U3 = R.

Let D be a nonzero derivation of a noncommutative prime ring R, and let U be the subring of R generated by all [D(x), x], x ∞ R. A classical theorem of Posner asserts that U is not contained in the center of R. Under the additional assumption that the characteristic of R is not 2, we prove a more general result stating that U contains a nonzero left ideal of R as well as a nonzero right ideal of R.