ISSN 2148-838X http://dx.doi.org/10.13069/jacodesmath.284954 J. Algebra Comb. Discrete Appl. 4(2) • 155–164 Received: 12 June 2015 Accepted: 20 February 2016 Journal of Algebra Combinatorics Discrete Structures and Applications Strongly nil ∗-clean rings Research Article Abdullah Harmancı, Huanyin Chen, A. Çiğdem Özcan Abstract: A *-ring R is called strongly nil *-clean if every element of R is the sum of a projection and a nilpotent element that commute with each other. In this paper we investigate some properties of strongly nil *- rings and prove that R is a strongly nil *-clean ring if and only if every idempotent in R is a projection, R is periodic, and R/J (R) is Boolean. We also prove that a *-ring R is commutative, strongly nil *-clean and every primary ideal is maximal if and only if every element of R is a projection. 2010 MSC: 16W10, 16U99 Keywords: Rings with involution, Strongly nil *-clean ring, *-Boolean ring, Boolean ring 1. Introduction Let R be an associative ring with unity. A ring R is called strongly nil clean if every element of R is the sum of an idempotent and a nilpotent that commute. These rings were first considered by Hirano-Tominaga-Yakub [9] and refered to as [E-N]-representable rings. In [7], Diesl introduces this class and studies their properties. The class of strongly nil clean rings lies between the class of Boolean rings and strongly π-regular rings (i.e. for every a ∈ R, a n ∈ Ra n+1 ∩ a n+1 R for some positive integer n) [7, Corollary 3.7]. An involution of a ring R is an operation ∗ : R → R such that (x + y) ∗ = x ∗ + y ∗ , (xy) ∗ = y ∗ x ∗ and (x ∗ ) ∗ = x for all x, y ∈ R. A ring R with an involution ∗ is called a ∗-ring. An element p in a ∗-ring R is called a projection if p 2 = p = p ∗ (see [2]). Recently the concept of strongly clean rings were considered for any ∗-ring. Vaš [12] calls a ∗-ring R strongly ∗-clean if each of its elements is the sum of a projection and a unit that commute with each other (see also [10]). In this paper, we adapt strongly nil cleanness to ∗-rings. We call a ∗-ring R strongly nil ∗-clean if every element of R is the sum of a projection and a nilpotent element that commute. The paper consists Abdullah Harmancı (Corresponding Author), A. Çiğdem Özcan; Hacettepe University, Department of Mathe- matics, 06800 Beytepe, Ankara, Turkey (email: harmanci@hacettepe.edu.tr, ozcan@hacettepe.edu.tr). Huanyin Chen; Hangzhou Normal University, Department of Mathematics, 310036, Hangzhou, China (email: huanyinchen@aliyun.com). 155