On strong lifting splits

An ideal I of a ring R is called enabling in R if for any a and e² = e ∈ R, the condition a-e ∈ I implies that a-f ∈ I for some f² = f ∈ Ra. If the above idempotent e is always chosen so that e = 1, then I is called weakly enabling in R In this note, a counterexample is given to show that I[x] need not be enabling in R[x] provided that the ideal I of a ring R is enabling in R, answering a question of M. Alkan, W.K. Nicholson, and A.C. Ozcan in the negative. Moreover we show that for any nil ideal I of a ring R, I[x] is enabling in R[x] if and only if I[x] is an exchange general ring if and only if I[x] is a clean general ring, and if and only if the Koethe´s Conjecture has a positive solution.