In this paper we study certain additive mappings Φ from a two-torsion free ring, R, with involution and 2R = R into a two-torsion free ring, A, which is prime (or semiprime) and 2A = A.

The first theorems assume that Φ(xx*) = Φ(x)Φ(x*) and conclude that Φ is a ring homomorphism or a Jordan homomorphism depending upon the conditions placed on the image ring. Similar results on derivations are also proved.

We also consider for prime rings with involution the same type of additive mappings on the symmetric elements as Small in her paper, "Mappings on simple rings with involution." We show that her results have obvious generalizations relative to the quotient ring of R.

Using these results, we obtain (for the Jordan modules in R which were studied by Jacobson and Small) analogous results relative to the questions of their being isomorphic.