Abstract
Firstly, the commutativity of rings is investigated in this paper. Let [Formula: see text] be a ring with identity. Then we obtain the following commutativity conditions: (1) if for each [Formula: see text] and each [Formula: see text], [Formula: see text] for [Formula: see text], where [Formula: see text] and [Formula: see text] are relatively prime positive integers, then [Formula: see text] is commutative; (2) if for each [Formula: see text] and each [Formula: see text], [Formula: see text] for [Formula: see text], where [Formula: see text] is a positive integer, then [Formula: see text] is commutative. Secondly, generalized 2-CN rings, a kind of ring being commutative to some extent, are investigated. Some relations between generalized 2-CN rings and other kinds of rings, such as reduced rings, regular rings, 2-good rings, and weakly Abel rings, are presented.
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