SPLIT-EXTENSIONS OF EXCHANGE RINGS: A DIRECT PROOF

Abstract. A direct ring-theoretic proof for ”if R is a ring, e is an idempotent and eRe and (1-e)R(1-e) are both exchange rings then R is also an exchange ring ” (left open for the last 31 years) is given. The exchange property was first (introduced and) studied for modules by Craw-ley P. and Jónsson B. in [1]: for a given cardinal m, a right R-module MR has the m-exchange property if whenever A = M ⊕ N = i∈