On Jordan and Jordan*-Generalized Derivations in Semiprime Rings with Involution
The purpose of this note is to prove the following result. Let R be a6 −torsion free semiprime *-ring and let G: R−→R be an additive mapping such that G(xyx )= G(x)y ∗ x ∗ + xD(y)x ∗ + xyD(x) holds for all x,y ∈ R and some *-derivations D of R. Then G is a Jordan*generalized derivation.
Jordan derivations on semiprime rings
I. N. Herstein has proved that any Jordan derivation on a 2- torsion free prime ring is a derivation. In this paper we prove that Herstein's result is true in 2-torsion free semiprime rings. This result makes it possible for us to prove that any linear Jordan derivation on a semisimple Banach algebra is continuous, which gives an affirmative answer to the question posed by A. M. Sinclair in (5). Preliminaries. Throughout this paper all rings will be associative. Let R be a ring. The center of R will be denoted by Z(R). We shall write (a, b) for ab — ba. A ring R is said to be 2-torsion free, if whenever 2a — 0, with a e R, then a = 0. A ring R is called a prime ring if aRb = (0) implies a = 0 or b = 0. A ring R is called a semiprime ring if aRa = (0) implies a = 0. Let R be any ring. An additive mapping ': R —y R is called a derivation if (ab)' = a'b + ab' holds for all pairs a,b e R. An additive mapping ': R —> R is called a Jordan derivation if (o2)' = a'a + aa' holds for all a e R. Obviously, every derivation is a Jordan derivation. The converse is, in general, not true. A well-known result of I. N. Herstein (2) states that every Jordan derivation on a 2-torsion free prime ring is a derivation. A brief proof of this result can be found in (1). The main purpose of this paper is to present a generalization of Herstein's result. More precisely, we shall prove that every Jordan derivation on a 2-torsion free semiprime ring is a derivation. In particular, every Jordan derivation on a 2-torsion free semisimple ring is a derivation, which generalizes a result of A. M. Sinclair (see (5)). From the fact that every linear derivation on a semisimple Banach algebra is continuous, and from our generalization of Herstein's result, it follows immediately that every Jordan derivation on a semisimple Banach algebra is continuous, which gives an affirmative answer to the question posed by A. M. Sinclair in (5). In the last part of the paper two characterizations of 2-torsion free prime rings are obtained.
Three-dimensional elasticity problems for the prismatic bar
A general solution is given to the three-dimensional linear elastic problem of a prismatic bar subjected to arbitrary tractions on its lateral surfaces, subject only to the restriction that they can be expanded as finite power series in the axial coordinate z. The solution is obtained by repeated differentiation of the tractions with respect to z, establishing a set of sub-problems . A recursive procedure is then developed for generating the solution to from that for . This procedure involves three steps: integration of the stress and displacement fields with respect to z, using an appropriate Papkovich–Neuber (P–N) representation; solution of two-dimensional in-plane and antiplane corrective problems for the tractions in that are independent of z; and expression of these corrective solutions in P–N form. The method is illustrated by an example.