An Idempotent-Based Generalization of Semicommutative Rings

We introduce and study several new classes of rings defined by idempotent conditions. A ring R is called i-semicommutative if (1−ab)aRb(1−ab)=0 whenever ab is a nonzero idempotent. This property lies strictly between semicommutativity and i-reversibility (where ab nonzero idempotent forces ba idempotent). We also define i-reduced rings (if a2 is a nonzero idempotent then a3=a) and i-domains (if ab is a nonzero idempotent then a∈I(b) or b∈I(a)). Basic properties are established, including closure under subrings, behaviour of corners, and connections with classical ring concepts. We characterize these properties for triangular matrix rings and for full matrix rings over commutative rings. Several examples illustrate the independence of the new notions from abelian, reversible and semicommutative rings. Open questions are posed concerning direct finiteness and matrix rings over noncommutative rings.