Journal of Pure and Applied Algebra 218 (2014) 661–665 Contents lists available at ScienceDirect Journal of Pure and Applied Algebra journal homepage: www.elsevier.com/locate/jpaa A note on completeness and strongly clean rings Alexander J. Diesl a , Thomas J. Dorsey b,∗ , Shelly Garg c,1 , Dinesh Khurana d a Department of Mathematics, Wellesley College, Wellesley, MA 02481, USA b Center for Communications Research, 4320 Westerra Court, San Diego, CA 92121-1969, USA c Department of Mathematics, Indian Institute of Science Education and Research, Mohali-140306, India d Department of Mathematics, Panjab University, Chandigarh-160014, India article info Article history: Received 31 January 2011 Received in revised form 15 July 2013 Available online 5 November 2013 Communicated by S. Iyengar MSC: 16U99; 16E50 abstract Many authors have investigated the behavior of strong cleanness under certain ring extensions. In this note, we investigate the classical problem of lifting idempotents, in order to consolidate and extend these results. Our main result is that if R is a ring which is complete with respect to an ideal I and if x is an element of R whose image in R/I is strongly π -regular, then x is strongly clean in R. This generalizes Theorem 2.1 of Chen and Zhou (2007) [9]. © 2013 Published by Elsevier B.V. In this note, all rings are associative and unital. Recall that, following [14], an element x of a ring R is said to be strongly clean if there is an idempotent e ∈ R, which commutes with x, such that x − e is a unit in R. If e is such an idempotent, we will say that x is e-strongly clean. Following [6], we will call x uniquely strongly clean if there is exactly one idempotent e such that x is e-strongly clean. As introduced in [1], an element x ∈ R is called strongly π -regular if there is a positive integer n such that x n ∈ x n+1 R ∩ Rx n+1 . Equivalently (see [4, Theorem 10.6]), the element x is strongly π -regular if and only if x satisfies Fitting’s Lemma as an endomorphism of R R . This implies that x is strongly π -regular if and only if there is an idempotent e ∈ R such that x is e-strongly clean and exe is nilpotent. If e is such an idempotent, we will say that x is e-strongly π -regular (in fact, such an idempotent is unique; see Lemma 2 below). A ring is said to be strongly clean (respectively, strongly π -regular) if each of its elements is strongly clean (respectively, strongly π -regular). There are many examples of, and results about, strongly clean rings which happen to be complete with respect to an ideal I (e.g. [7, Theorem 2.4], [8, Theorem 9], [16, Theorem 2.7], [12, Theorem 2.10], and [3, Theorem 25, Corollary 26]). The aforementioned results demonstrate that it is frequently true (though not always, as seen in [2, Example 45]) that, if R/I is strongly clean and R is I -adically complete, then R will be strongly clean. Moreover, the use of completeness in this context often simplifies proofs greatly (e.g. see [3, Theorem 25, Corollary 26] and the surrounding discussion). This note continues that theme. The present investigation is motivated by [9, Theorem 2.1] (see also [15, Theorem 3] for a result in the nonunital case and [6, Theorem 20] for a uniqueness statement) which states that for a ring R and a ring endomorphism σ of R, an element of the skew power series ring R[[x; σ ]] is strongly clean provided that its constant term is strongly π -regular in R. The proof given there is rather long and technical, and makes critical use of the added structure present in the power series ring. In this note, we aim to both simplify and generalize this result. The first of our two main theorems is a direct generalization of the above-mentioned results with a streamlined proof that clearly highlights the role that completeness plays in the argument. Our second theorem offers an alternate approach, which serves two purposes. First, it illustrates an interesting method for ∗ Corresponding author. E-mail addresses: adiesl@wellesley.edu (A.J. Diesl), dorsey@ccrwest.org (T.J. Dorsey), shellygarg24@gmail.com (S. Garg), dkhurana@pu.ac.in (D. Khurana). 1 A portion of this work constitutes part of the Ph.D. dissertation of the third author. 0022-4049/$ – see front matter © 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.jpaa.2013.08.006