A nonzero ring R is called a factorization ring (or F-ring) if each nonzero element of R can be expressed as a product of two (not necessarily distinct) elements. We give conditions for an F-ring to have 1. We discuss subrings of F-rings, and show that certain F-rings must be finite. For finite F-rings R, we study the number  defined to be the maximum number of 2-factorizations a nonzero element of R can possess.

factorization, finiteness.