martindale proved that under some conditions every multiplicative isomorphism between two rings is additive. in this paper, we extend this theorem to a larger class of mappings and conclude that every multiplicative $(alpha, beta)-$derivation is additive.

Martindale proved that under some conditions every multiplicative isomorphism between two rings is additive. In this paper, we extend this theorem to a larger class of mappings and conclude that every multiplicative $(alpha, beta)-$derivation is additive.

let be a set and let be the set of subsets of . the pair in which is a collection of elements of (blocks) is called a design if every element of appears in , times. is called a symmetric design if . in a symmetric design, each element of appears times in blocks of . a mapping between two designs and is an isomorphism if is a one-to-one correspondence and . every isomorphism of a design, , to it...

2 Basic group theory 2 2.1 The definition of a group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Group homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.3 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.4 Cosets and quotient spaces . . . . . . . . . . . . . . . . . . . . . . . . . . ....