Strong Lifting Splits

The concept of an enabling ideal is introduced so that an ideal I is strongly lifting if and only if it is lifting and enabling. These ideals are studied and their properties are described. It is shown that a left duo ring is an exchange ring if and only if every ideal is enabling, that Zhous -ideal is always enabling, and that the right singular ideal is enabling if and only if it is contained in the Jacobson radical. The notion of a weakly enabling left ideal is de\u85ned, and it is shown that a ring is an exchange ring if and only if every left ideal is weakly enabling. Two related conditions, interesting in themselves, are investigated: The \u85rst gives a new characterization of -small left ideals, and the second characterizes weakly enabling left ideals. As an application (which motivated the paper), let M be an I-semiregular left module where I is an enabling ideal. It is shown that m 2 M is I-semiregular if and only if m q 2 IM for some regular element q of M and, as a consequence, that if M is countably generated and IM is -small in M; then M = 1i=1Rei where e2i = ei 2 R for each i: Throughout this paper all rings are associative with unity, and all modules are unitary. We denote the Jacobson radical of a ring R by J = J(R); the right and left socles by Sr and Sl; the right and left singular ideals by Zr and Zl; and the left -ideal [8] by = (R): We write I C R to indicate that I is an ideal (right and left) of R: The ring of integers is denoted by Z; and we write Zn for the ring of integers modulo n: If X is a subset of R we write r(X) and l(X) for the right and left annihilators of X in R: Modules are left modules unless otherwise indicated, and homomorphisms are written on the right. The notation S M indicates that S is a direct summand of M:1 1. Enabling Ideals Let I be an ideal of a ring R: Then I is called lifting if, for each a 2 R; the condition a2 a 2 I implies that a e 2 I for some e2 = e 2 R: In [6], the ideal I is called strongly lifting if for eac