Clean Elements in Commutative Reduced Rings#

Abstract

As defined by Nicholson [Nicholson, W. K. (1977). Lifting idempotents and exchange rings. Trans. Amer. Math. Soc. 229:269–278] an element of a ring R is clean if it is the sum of a unit and an idempotent, and a subset A of R is clean if every element of A is clean. It is shown that a semiprimitive Gelfand ring R is clean if and only if Max(R) is zero-dimensional; if and only if for each M ∈ Max(R), the intersection all prime ideals contained in M is generated by a set of idempotents. We also give several equivalent conditions for clean functional rings. In fact, a functional ring R is clean if and only if the set of clean elements is closed under sum; if and only if every zero-divisor is clean; if and only if; R has a clean prime ideal.