On strong commutativity-preserving maps

Let R be a ring with center Z(R). We write the commutator [x, y] = xy− yx, (x, y ∈ R). The following commutator identities hold: [xy,z] = x[y,z] + [x,z]y; [x, yz] = y[x,z] + [x, y]z for all x, y,z ∈ R. We recall that R is prime if aRb = (0) implies that a= 0 or b = 0; it is semiprime if aRa = (0) implies that a = 0. A prime ring is clearly a semiprime ring. A mapping f : R→ R is called centralizing if [ f (x),x] ∈ Z(R) for all x ∈ R; in particular if [ f (x),x] = 0 for all x ∈ R, then it is called commuting. A commuting map is centralizing but the converse is not true, in general. It is easy to see that if f : R→ R is an additive and commuting map, then [ f (x), y] = [x, f (y)] for all x, y ∈ R. A mapping f : R→ R is called commutativity preserving if [ f (x), f (y)] = 0 whenever [x, y] = 0. Commutativity-preserving maps have been extensively studied on operator algebras (see [7, 9, 11, 12, 13] and the references therein). Many authors have also worked on commutativity-preserving maps on rings (see [1, 2, 6, 8], where further references are also given). There has also been considerable interest in strong commutativity-preserving maps. A mapping f : R→ R is called strong commutativity preserving if [ f (x), f (y)] = [x, y] for all x, y ∈ R. A strong commutativity-preserving map is commutativity preserving but the converse does not hold, in general. We recall that an additive map f from a ring R into itself is called an antihomomorphism if f (xy) = f (y) f (x) for all x, y ∈ R. We will follow Herstein [10] for other undefined notations and terminology used here. In this paper, we mainly study commutativity-preserving and strong commutativitypreserving properties of homomorphisms and antihomomorphisms of certain rings. We