W. K. Nicholson

Lifting idempotents and exchange rings

Idempotents can be lifted modulo a one-sided ideal L of a ring R if, given x e R with x-x2 cL, there exists an idempotent e c 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 S N and B C M. In 1972 Warfield showed that if M is a module over an associative ring R 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 Jonsson) that every projective module over an exchange ring is a direct sum of cyclic submodules. Let J(R) denote the Jacobson radical of R. Warfield showed that, if R/J(R) is (von Neumann) regular and idempotents can be lifted modulo J(R), then R is an exchange ring and so generalized theorems of Kaplansky and Muller. The main purpose of this paper is to prove the following theorem: A ring R is an exchange ring if and only if idempotents can be lifted modulo every left (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. 1. Suitable rings. In this section, the rings of interest are defined, some of their properties are deduced, and several examples are given. All rings are assumed to be associative with identity and J(R) denotes the Jacobson radical of a ring R. 1.1. PROPOSITION. If R is a ring, the following conditions are equivalent for an element x of R. Received by the editors December 2, 1975. AMS (MOS) subject classifications (1970). Primary 16A32, 16A64; Secondary 16A30, 16A50.

Strongly clean rings and fitting's lemma

A ring is called strongly clean if every element is the sum of an idempotent and a unit which commute. These rings are shown to be a natural generalization of the strongly π-regular rings, and several properties of strongly π-regular rings are extended, including their relationship to Fittings lemma.

EXTENSIONS OF CLEAN RINGS

It is shown that if e is an idempotent in a ring R such that both eRe and (1 − e)R(1 − e) are clean rings, then R is a clean ring. This implies that the matrix ring M n (R) over a clean ring is clean, and it gives a quick proof that every semiperfect is clean. Other extensions of clean rings are studied, including group rings.

Semiregular modules and rings

Mares [ 9 ] has called a projective module semiperfect if every homomorphic image has a projective cover and has shown that many of the properties of semiperfect rings can be extended to these modules. More recently Zelmanowitz [ 16 ] has called a module regular if every finitely generated submodule is a projective direct summand. In the present paper a class of semiregular modules is introduced which contains all regular and all semiperfect modules. Several characterizations of these modules are given and a structure theorem is proved. In addition several theorems about regular and semiperfect modules are extended.

RINGS IN WHICH ELEMENTS ARE UNIQUELY THE SUM OF AN IDEMPOTENT AND A UNIT

An associative ring with unity is called clean if every element is the sum of an idempotent and a unit; if this representation is unique for every element, we call the ring uniquely clean. These rings represent a natural generalization of the boolean rings in that a ring is uniquely clean if and only if it is boolean modulo the Jacobson radical and idempotents lift uniquely modulo the radical. We also show that every image of a uniquely clean ring is uniquely clean, and construct several noncommutative examples.

Countable linear transformations are clean

It is shown that every linear transformation on a vector space of countable dimension is the sum of a unit and an idempotent. An element in a ring R is called clean in R if it is the sum of a unit and an idempotent, and the ring itself is called clean if every element is clean. Every clean ring is an exchange ring and, if R has central idempotents, R is an exchange ring if and only if it is clean [2, Proposition 1.8]. Camillo and Yu [1, Theorem 9] have shown that a ring is semiperfect if and only if it is clean and has no infinite orthogonal family of idempotents. Our main result is the following theorem which answers a question of P. Ara. Theorem. If VD is a vector space of countably infinite dimension over a division ring D, then end(VD) is clean. A ring R is called unit regular if, for each a E R, there exists a unit u E R such that aua = a. Camillo and Yu [1, Theorem 5] show that every unit regular ring is clean. The Theorem shows that the converse is not true. Corollary. There exists a (von Neumann) regular, right self-injective, clean ring which is not unit regular. Proof. The ring end(VD) in the Theorem suffices because it is not unit regular. In fact, it is not even Dedekind finite (ab= 1 implies ba= 1). D The proof of the Theorem employs several preliminary lemmas. Throughout this paper D always denotes a division ring and VD is always a vector space of countably infinite dimension over D. If {xI, x2, ... } is a basis of VD, the linear transformation a: V -* V given by a(xi) = xi+, for each i is called a shift operator on V. Lemma 1. Every shift operator on VD is clean in end(VD). Received by the editors July 16, 1996. 1991 Mathematics Subject Classification. Primary 16S50; Secondary 16E50, 16U99.

Principally quasi-injective modules

An R-module M is called principally quasi-injective if each R-hornomorphism from a principal submodule of M to M can be extended to an endomorphism of M. Many properties of principally injective rings and quasi-injective modules are extended to these modules. As one application, we show that, for a finite-dimensional quasi-injective module M in which every maximal uniform submodule is fully invariant, there is a bijection between the set of indecomposable summands of M and the maximal left ideals of the endomorphism ring of M Throughout this paper all rings R are associative with unity, and all modules are unital. We denote the Jacobson radical, the socle and the singular submodule of a module M by J(M), soc(M) and Z(M), respectively, and we write J(M) = J. The notation N ⊆ess M means that N is an essential submodule of M.

WEAKLY CONTINUOUS AND C2-RINGS

A ring R is called right weakly continuous if the right annihilator of each element is essential in a summand of R, and R satisfies the right C2-condition (every right ideal that is isomorphic to a direct summand of R is itself a direct summand). We show that a ring R is right weakly continuous if and only if it is semiregular and J(R) = Z(R R ). Unlike right continuous rings, these right weakly continuous rings form a Morita invariant class. The rings satisfying the right C2-condition are studied and used to investigate two conjectures about strongly right Johns rings and right FGF-rings and their relation to quasi-Frobenius rings.

On exchange rings

A short, elementary proof is given that right exchange rings are left exchange, with an application to exchange rings with one in the stable range.

Normal radicals and normal classes of rings

Abstract A radical N in the category of rings is called normal if, for any Morita context (R, V, W, S), we have VN(S)W ⊆ N(R). In this paper these radicals are investigated and the related notion of a normal class of prime rings is defined. A characterization of normal, special radicals is given and it is shown that normal classes generalize special classes in a natural way. Several results are given on the closure of normal classes under forming related rings and some theorems on structure spaces are extended.