Semifine Rings: A Generalization of Fine Rings

A ring R is called semifine if every nonzero element can be written as the sum of a full element (an element f with RfR = R) and a nilpotent element. This paper introduces and studies semifine rings, a natural generalization of fine rings in which a full element replaces the unit. Every fine ring is semifine, but the converse is false. We prove that the ring of n × n matrices over the integers, M n (ℤ), is semifine for all n ≥ 2, providing the first non-simple example. We give a complete elementary description of fullness in M n (ℤ): a matrix is full if and only if the greatest common divisor of its entries is 1. We show that semifineness is not preserved under Morita equivalence and that corner rings of semifine rings need not be semifine. We prove that matrix rings over semifine rings are semifine, and that power series rings over semifine rings containing a full nilpotent are also semifine.