A new study of strongly clean elements in rings

The principal goal of this paper is to make some new contributions to the very active study of strongly clean elements in rings. Starting with a general Equivalence Theorem in Sec. 2 characterizing strongly clean elements in an arbitrary ring, the paper goes on in Sec. 3 to offer some examples and applications of such elements. After this, uniquely strongly clean and double-centralizer clean elements are introduced and studied in Sec. 4, culminating in a final Sec. 5 with six open questions on strongly clean elements and strongly clean rings.