Neat rings

A ring is called clean if every element is the sum of a unit and an idempotent. Throughout the last 30 years several characterizations of commutative clean rings have been given. We have compiled a thorough list, including some new equivalences, in hopes that in the future there will be a better understanding of this interesting class of rings. One of the fundamental properties of clean rings is that every homomorphic image of a clean ring is clean. We define a neat ring to be one for which every proper homomorphic image is clean. In particular, the ring of integers, Z, and any nonlocal PID are examples neat rings which are not clean. We characterize neat Bézout domains using the group of divisibility. In particular, it is shown that a neat Bézout domain has stranded primes, that is, for every nonzero prime ideal the set of primes either containing or contained in the given prime forms a chain under set-theoretic inclusion. © 2005 Elsevier B.V. All rights reserved. MSC: Primary: 13F99; secondary: 06F20 1. A history of clean rings In this section, we give an account of the class of clean rings. Over the past 25 years many authors in several different contexts have investigated clean rings. Our focus is on commutative clean rings. For a detailed reference on not necessarily commutative clean rings and exchange rings, see [29]. In Theorem 1.7, we give a list of characterizations of commutative clean rings. Most are old theorems but some new ones are also included. The theorem includes a collage of different kinds of rings. For the sake of completeness, we E-mail address: warrenb@bgnet.bgsu.edu. 0022-4049/$ see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jpaa.2005.07.012 244 W.Wm. McGovern / Journal of Pure and Applied Algebra 205 (2006) 243–265 shall define most of the necessary concepts. To that end, we begin with the definition of a clean ring. The ring A is called clean if every element is the sum of an idempotent and a unit. If every element of A is clean, then we say A is a clean ring. Some examples of clean rings include all commutative von Neumann regular rings, all local rings, any ring Mn(R) of n × n matrices over a clean ring, and semiperfect rings. Furthermore, the class of clean rings is closed under products and homomorphic images. The definition of clean ring is due to Nicholson [27] and was shown to be a strengthening of the next condition which is due to Crawly and Jónsson [10] and, then later, Warfield [31]. An A-module M is said to have the finite exchange property if for any module N and decompositions N = M ′ ⊕ P = ⊕ i∈I Qi , where M ′ M and I is finite, then there exist submodules Q′i ⊆ Qi , for each i, such that N = M ′ ⊕ (⊕Q′i ). A is called an exchange ring if it is a finite exchange A-module. As mentioned above Nicholson showed that a clean ring is an exchange ring and when idempotents are central in A the reverse holds. In general, there is an example of an exchange ring which is not clean (see [6]). Example 1.1. As mentioned above the class of clean rings is closed under arbitrary products and homomorphic images. Furthermore, it is known that every commutative zerodimensional ring, and hence boolean rings, are clean. An integral domain is clean precisely when it is local. A clean ring is a Gelfand ring. Recall that a ring is called a Gelfand ring if whenever a + b = 1 there are r, s ∈ A such that (1 + ar)(1 + bs) = 0. A ring is called a pm-ring if every prime ideal is contained in a unique maximal ideal. Commutative pm-rings were first studied in [13] and later in [9]. It is known that for any topological space X the ring C(X) consisting of all real-valued continuous functions on X under the pointwise operations is always a pm-ring. It had been asserted that a commutative ring is a Gelfand ring if and only if it is a pm-ring. Recently, Banaschewski [3] has shown that whether this is true actually depends on one’s set theoretic axioms. In particular, every commutative pm-ring is a Gelfand ring if and only if the Prime Ideal Theorem holds. In this present article, we shall not delve into set theoretic matters and simply work within the confines of ZFC. For a ring A, Spec(A) denotes the collection of prime ideals of A. The hull-kernel topology (or Zariski topology) on Spec(A) is the topology obtained by taking the collection of sets of the form U(a) = {P ∈ Spec(A) : a / ∈ P } for arbitrary a ∈ A as a base for the open sets. Observe that U(a) ∩ U(b) = U(ab). We write V(a) for the complement of U(a). The collection of maximal ideals of A is denoted W.Wm. McGovern / Journal of Pure and Applied Algebra 205 (2006) 243–265 245 by Max(A). The hull-kernel topology on Max(A) is simply the subspace topology Max(A) inherits from the hull-kernel topology on Spec(A). In particular, we let U(a) = Max(A) ∩ U(a)