Two-Sided Properties of Elements in Exchange Rings Dinesh Khurana, T

Two-Sided Properties of Elements in Exchange Rings Dinesh Khurana, T. Y. Lam, and Pace P. Nielsen Abstract For any element a in an exchange ring R, we show that there is an idempotent e 2 aR \ R a such that 1 − e 2 (1 − a) R \ R (1 − a). A closely related result is that a ring R is an exchange ring if and only if, for every a 2 R, there exists an idempotent e 2 R a such that 1 − e 2 (1 − a) R. The Main Theorem of this paper is a general two-sided statement on exchange elements in arbitrary rings which subsumes both of these results. Finally, applications of these results are given to the study of the endomorphism rings of exchange modules. x1. Introduction Throughout this paper, R denotes a unital (generally noncommutative) ring. Following [Ni1], we say that an element a 2 R is right suitable if it has the following \idempotent lifting" property: for any right ideal I ⊆ R containing a−a2, there exists an idempotent e 2 R such that e − a 2 I. The set of right suitable elements in R is denoted by suitr(R), and the set suit`(R) of left suitable elements in R is defined similarly. In [Ni2], Nicholson proved that suit`(R) = suitr(R) for any ring R. The rings R with the property that R = suitr(R) are called (right) suitable rings. In the seminal paper [Ni1], Nicholson proved that these are precisely Warfield’s exchange rings in [Wa]; namely, those rings R for which the right module RR satisfies the (finite) exchange property of Crawley and Jónsson. In our paper, we'll use the terms \suitable rings" and \exchange rings" interchangeably, noting that suitable elements are sometimes also called \exchange elements" in the literature. Various characterizations for right suitable elements in rings were given in [Ni1]. Among the most useful ones is the following \Goodearl-Nicholson characterization" (see [GW] and [Ni1]): a 2 suitr(R) if and only if there exists an idempotent e 2 aR such that 1 − e 2 (1 − a) R. (In particular, R is an exchange ring if and only if every element a 2 R has the above idempotent property.) However, the prevailing study of the suitability of elements in rings has been mostly carried out \on one side" of the ring at a time. To our best knowledge, there has been almost no work in which the existence of idempotents with \two-sided" (left and right) properties was investigated. Our main results in this paper are intended to be the beginning steps in this direction. For ease of understanding, we state explicitly below two special cases of the Main Theorem in this paper. 1 Theorem A. For every element a in an exchange ring R, there exists an idempotent e 2 aR a such that 1 − e 2 (1 − a) R (1 − a). Note that this Theorem A is in fact equivalent to the first result stated in the Abstract. One simply observes that, if an idempotent e lies in an intersection xR \ R y, then e 2 xR y. Indeed, writing e = xr = sy for some r; s 2 R, we have e = e2 = (xr)(sy) 2 xR y. And of course, xR y ⊆ xR \ R y for any x; y 2 R. Theorem A appears to be new even for a von Neumann regular ring (or a π-regular ring) R, as we have not been able to locate such a result in the standard treatises on the subject (e.g. [vN1], [vN2], or [Go]). The same result also seems to be unknown for semiperfect rings and C∗-algebras of real rank zero, both of which are well known examples of exchange rings (by [Wa] and [AG]). In the case where R is an exchange ring of stable range one, a result of Ara given in [Ni2: Corollary] states that, for every a 2 R, there exists a pair of conjugate idempotents e; e0 such that e 2 aR; 1−e 2 (1−a)R; e0 2 R a, and 1 − e0 2 R (1 − a). A version of this result for suitable elements under the assumption that R satisfies internal cancellation appeared in [KG: Proposition 4.2]. Theorem A (or more precisely, Theorem 3.6 in x3) is a substantial improvement of both of these results in that the stable range one assumption and the internal cancellation assumption on R have been dropped, and the conclusion is strengthened to yield a single idempotent e = e0 with the aforementioned properties. The second result we want to specifically mention in this Introduction is the following \left-right mixed" characterization for an exchange ring, for which we also know of no explicit reference in the literature. Theorem B. A ring R is an exchange ring if and only if, for every a 2 R, there exists an idempotent e 2 R a such that 1 − e 2 (1 − a) R. For the larger part of this paper, instead of working with exchange (or suitable) rings, we take the more flexible approach of working with suitable elements. From this viewpoint, most of our results are stated in, and applicable to, a more general \element-wise" setting. In particular, Theorem A and Theorem B above are just special cases of the Main Theorem in this paper, which is stated as Theorem 2.5 in x2. Some ramifications and applications of this Main Theorem are given in x3. The paper closes with a final section (x4), in which the two-sided properties obtained for suitable elements in x3 are applied to the study of the endomorphism rings of modules. In some sense, the decomposition results for endomorphisms obtained in x4 are generalizations of the \Fitting decompositions" in Fitting's classical paper [Fi]. The terminology and notations introduced so far in this Introduction will be used freely throughout the paper. In addition, the following useful definitions are needed in xx3-4: a ring element a 2 R is said to be clean if a = f + u for some idempotent f 2 R and some unit u 2 R, and a is said to be strongly clean if there exists 2 such a decomposition a = f + u with fu = uf. It is well known (from the proof of [Ni1: Proposition 1.8]) that clean elements are suitable. Next, an element a 2 R is said to be strongly π-regular if there exists an integer n ≥ 1 such that an 2 an+1R \ R an+1. According to Nicholson [Ni3: Theorem 1], strongly π-regular elements are strongly clean. As in our companion paper [KLN], we'll denote the set of idempotents in a ring R by idem (R). Other standard terminology and conventions in ring theory follow those in [AF], [Go], and [La]. Whenever it is more convenient, we'll use the widely accepted shorthand “iff” for \if and only if" in the text. x2. Main Theorem and Its Proof The goal of this section is to prove the Main Theorem 2.5 below on the existence of idempotents with certain two-sided properties arising from a \commuting unimodular equation" pq + st = 1 in a ring. We start with a lemma that will be crucial for the proof of Theorem 2.5. Lemma 2.1. (A) Let p; q; c; d be commuting elements in R. If pq + qcxd 2 idem (R) for some x 2 R, then (pq)2 + cydq 2 idem (R) for some y 2 R. (B) Let p; q; s; t be commuting elements of R with pq + st = 1. If there exists an idempotent e 2 pRq such that 1 − e 2 sR t, then there exists an idempotent g 2 pqR such that 1 − g 2 sR t. Proof. A basic fact to be used several times in this proof is that, in any ring R, ab 2 idem (R) ) (ba)2 2 idem (R). This is checked by simply noting that (ba)4 = b (ab)3 a = b (ab) a = (ba)2. (A) Since q (p + cxd) 2 idem (R), we have (pq + cxdq)2 2 idem (R). By inspection, (pq + cxdq)2 2 (pq)2 + cRdq. (B) To facilitate the proof of (B), we first observe that, for any integer n ≥ 1, (2:2) 1 − (pq)n = (1 − pq) (1 + ··· + (pq)n−1) 2 sR t: Write the given idempotent e in (B) in the form p (1 + x) q for some x 2 R. Since 1 − e = 1 − pq − pxq = st − pxq 2 sR t, we have pxq = syt for some y 2 R. Thus, (2:3) q (syt) = q (pxq) = (1 − st) xq ) xq 2 sR: Similarly, (syt) p = (pxq) p = px (1 − st) ) px 2 R t. As e = p (1 + x) q is an idempotent, so is (2:4) h:= (qp (1 + x))2 = (qp)2 + qpqpx + qpxqp + qpxqpx 2 pqR: In view of px 2 R t and (2.2) (for n = 2), we have 1−h 2 R t. Similarly, using xq 2 sR and (2.2) (for n = 3), we have 1 − hqp 2 sR. Now let g := h + h (qp) (1 − h), which is easily seen to be an idempotent by a direct calculation. This is the idempotent we want, since g 2 pqR by (2.4), and we have also 1 − g = (1 − h) − h (qp) (1 − h) = (1 − hqp) (1 − h) 2 (sR)(R t) = sR t: 3 We can now state and prove the following main result of this paper.

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