Commun. Korean Math. Soc. 32 (2017), No. 2, pp. 267–276 https://doi.org/10.4134/CKMS.c160102 pISSN: 1225-1763 / eISSN: 2234-3024 RINGS WITH MANY REGULAR ELEMENTS Nahid Ashrafi and Ebrahim Nasibi Abstract. In this paper we introduce rings that satisfy regular 1-stable range. These rings are left-right symmetric and are generalizations of unit 1-stable range. We investigate characterizations of these kind of rings and show that these rings are closed under matrix rings and Morita Context rings. 1. Introduction Let R be an associative ring with an identity. We say that R has stable range one provided that aR + bR = R with a,b ∈ R implies that there exists some y ∈ R such that a + by ∈ U (R), where U (R) denotes the set of all units in R. One of the most important features of stable range one is the cancellation of related modules from direct sums. Evans [15, Theorem 2] proved that if A,B,C are R-modules such that A ⊕ B ∼ = A ⊕ C, and End R (A) has stable range one, then B ∼ = C. Stable range conditions have been studied in [1], [8], [9], [11], [14], [19] and [21]. Goodearl and Mental [16] defined the concept of unit 1-stable range: we say that R satisfies unit 1-stable range provided that for any a,b ∈ R, aR + bR = R implies there exists a y ∈ U (R) such that a + by ∈ U (R). Many authors have studied this class of rings such as [7], [12], [13] and [16]. Here we generalize this concept as bellow. Definition 1.1. A ring R is said to satisfy regular 1-stable range provided that for any a,b ∈ R, aR + bR = R implies there exists a regular (von Neumann) element r ∈ R such that a + br ∈ U (R). Obviously, if R satisfies unit 1-stable range, then it satisfies regular 1-stable range. But the converse is not true in generally. For example, Z/2Z (the ring of integers modulo 2) satisfies regular 1-stable range, while it does not satisfies unit 1-stable range. In this paper, we will prove that a ring satisfies regular 1-stable range is left-right symmetric. In other words, a ring R satisfies regular 1-stable range Received May 4, 2016; Revised June 28, 2016. 2010 Mathematics Subject Classification. 16E50, 16U99. Key words and phrases. stable range one, unit 1-stable range, regular 1-stable range. c 2017 Korean Mathematical Society 267