Exchange rings, exchange equations, and lifting properties

In this paper, we study exchange rings and clean rings R with 2 is an element of U(R) (or otherwise). Analogues of a theorem of Camillo and Yu characterizing clean and strongly clean rings with 2 is an element of U(R) are obtained for such rings (as well as for exchange rings) using the viewpoint of exchange equations introduced in a recent paper of the authors. We also study a new class of rings including von Neumann regular rings in which square roots of one (instead of idempotents) can be lifted modulo left ideals, and conjecture that such rings are exchange rings. This conjecture holds for commutative rings, and would hold for all rings if it holds for semiprimitive rings of characteristic 2.