Abstract.Idempotents can be lifted modulo a one-sided ideal L of a ring R if, given x E R with x -x2 6 L, there exists an idempotent e S R such that e -x e L. Rings in which idempotents can be lifted modulo every left (equivalently right) ideal are studied and are shown to coincide with the exchange rings of Warfield. Some results of Warfield are deduced and it is shown that a projective module P has the finite exchange property if and only if, whenever P = N + M where N and M are submodules, there is a decomposition P = A © B with A C. N and B Q M.In 1972 Warfield showed that if M is a module over an associative ring 7? then M has the finite exchange property if and only if end M has the exchange property as a module over itself. He called these latter rings exchange rings and showed (using a deep theorem of Crawley and Jónsson) that every projective module over an exchange ring is a direct sum of cyclic submodules.Let /(7?) denote the Jacobson radical of R. Warfield showed that, if R/J(R) is (von Neumann) regular and idempotents can be lifted modulo /(7?), then 7? is an exchange ring and so generalized theorems of Kaplansky and Müller.The main purpose of this paper is to prove the following theorem: A ring R is an exchange ring ¡S and only // idempotents can be USted modulo every leSt (respectively right) ideal. The properties of these rings are examined in the first section and the theorem is proved in the second section. The theorems of Warfield are then easily deduced and a new condition that a projective module have the finite exchange property is given.

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Abstract.Idempotents can be lifted modulo a one-sided ideal L of a ring R if, given x E R with x -x2 6 L, there exists an idempotent e S R such that e -x e L. Rings in which idempotents can be lifted modulo every left (equivalently right) ideal are studied and are shown to coincide with the exchange rings of Warfield. Some results of Warfield are deduced and it is shown that a projective module P has the finite exchange property if and only if, whenever P = N + M where N and M are submodules, there is a decomposition P = A © B with A C. N and B Q M.In 1972 Warfield showed that if M is a module over an associative ring 7? then M has the finite exchange property if and only if end M has the exchange property as a module over itself. He called these latter rings exchange rings and showed (using a deep theorem of Crawley and Jónsson) that every projective module over an exchange ring is a direct sum of cyclic submodules.Let /(7?) denote the Jacobson radical of R. Warfield showed that, if R/J(R) is (von Neumann) regular and idempotents can be lifted modulo /(7?), then 7? is an exchange ring and so generalized theorems of Kaplansky and Müller.The main purpose of this paper is to prove the following theorem: A ring R is an exchange ring ¡S and only // idempotents can be USted modulo every leSt (respectively right) ideal. The properties of these rings are examined in the first section and the theorem is proved in the second section. The theorems of Warfield are then easily deduced and a new condition that a projective module have the finite exchange property is given.

A ring R satisfies the dual of the isomorphism theorem if R/Ra ∼ = l(a) for all elements a of R, where l(a) denotes the left annihilator. We call these rings left morphic. Examples include all unit regular rings and certain left uniserial local rings. We show that every left morphic ring is right principally injective, and use this to characterize the left perfect, right and left morphic rings.  2004 Elsevier Inc. All rights reserved.It is a well-known theorem of Erlich [5] that a map α in the endomorphism ring of a module M is unit regular if and only if it is regular and M/im(α) ∼ = ker(α). Our focus is on the case M = R R, so if α = · a : R R → R R is right multiplication by the element a ∈ R, the condition becomes R/Ra ∼ = l(a) where l(a) denotes the left annihilator. We say that the ring R is left morphic if every element satisfies this condition. We begin (Theorem 9) by characterizing the left morphic, local rings with nilpotent radical (and call these rings left 'special'); in particular we show that these rings are all left artinian. We show (Theorem 29) that a semiperfect left morphic ring is a finite product of matrix rings over local left morphic rings, we use this result to characterize (in Theorem 35) the left perfect, left and right morphic rings as the finite products of matrix rings over left and right 'special' rings.Along the way, we show (Theorem 24) that every left morphic ring is right principally injective [11]. With this we see that the left morphic rings R with ACC on right annihilators are left artinian, and have the property that eRe is left 'special' for every local idempotent e