Additivity of multiplicative (generalized) maps over rings
Abstract
In this paper, we mainly prove some results on the additivity of maps over rings under certain conditions. First, we discuss a special case of MARTINDALE III's theorem of 1969M as a bijective map over a ring R with a non-trivial idempotent satisfying (ab)=(a)(b) for all a, b∈ R, is additive. Then we prove that a map D on R satisfying D(ab)=D(a)b+(a) D(b) for all a,b∈ R, where is the map mentioned above, is additive. Finally, we establish that if a map g over R satisfies g(ab)=g(a)b+(a)D(b), for all a,b∈ R and the maps and D are mentioned above, then g is additive.
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