Semisimple module

Abstract: 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.

Abstract: Injective modules.- Essential extensions and the injective hull.- Quasi-Injective modules.- Radical and semiprimitivity in rings.- The endomorphism ring of a quasi-injective module.- Noetherian, artinian, and semisimple modules and rings.- Rational extensions and lattices of closed submodules.- Maximal quotient rings.- Semiprime rings with maximum condition.- Nil and singular ideals under maximum conditions.- Structure of noetherian prime rings.- Maximal quotient rings.- Quotient rings and direct products of full linear rings.- Johnson rings.- Open problems.