On Generalizations of Commutativity
This note is concerned with generalizations of commutativity. We introduce identity symmetric and right near commutative, and study basic structures of rings with such ring properties. It is shown that if R is an identity symmetric ring, then the set of all nilpotent elements forms a commutative subring of R. Moreover, identity symmetric regular rings are proved to be commutative. The near commutativity is shown to be not left right symmetric, and we study some conditions under which the near commutativity is left right symmetric. We also examine the near commutativity of skew trivial extensions, which has a role in this note.