RINGS IN WHICH ELEMENTS ARE UNIQUELY THE SUM OF AN IDEMPOTENT AND A UNIT
Glasgow Mathematical Journal
An associative ring with unity is called clean if every element is the sum of an idempotent and a unit; if this representation is unique for every element, we call the ring uniquely clean. These rings represent a natural generalization of the Boolean rings in that a ring is uniquely clean if and only if it is Boolean modulo the Jacobson radical and idempotents lift uniquely modulo the radical. We also show that every image of a uniquely clean ring is uniquely clean, and construct several
... ruct several noncommutative examples. 2000 Mathematics Subject Classification. Primary 16E50, secondary 16U99, 16S70. 1. Introduction. An element a in a ring R is called clean if a is the sum of an idempotent and a unit in R, and R is called a clean ring if every element is clean. It is known 4, Proposition 1.8 that clean rings are exchange rings (see Warfield 6) and that the two concepts are equivalent for rings with all idempotents central. Moreover Camillo and Yu have shown that every unit regular ring is clean 2, Theorem 5, and that the clean rings with no infinite orthogonal families of idempotents are precisely the semiperfect rings 2, Theorem 9. In this paper we investigate the uniquely clean rings in which each element has a unique representation as the sum of an idempotent and a unit, and find that these rings are closely related to the Boolean rings. In fact, we prove the following results. THEOREM. A local ring R is uniquely clean if and only if R/J(R) THEOREM. A ring R is uniquely clean if and only if R/J(R) is Boolean and idempotents lift uniquely modulo J(R). In particular R is Boolean if and only if R is uniquely clean and J(R) THEOREM. Every image of a uniquely clean ring is again uniquely clean. We also use ideal extensions to construct several examples of uniquely clean rings, some of which are not commutative. Throughout this paper all rings are associative with unity (unless otherwise noted) and all modules are unitary. We denote the group of units of the ring R by U = U(R), the center by C(R), and the Jacobson radical by J = J(R), and we write I R to https://www.cambridge.org/core/terms. https://doi. In particular, no matrix ring M n (R), and no triangular matrix ring T n (R), is uniquely clean if n ≥ 2. Example 3 and Lemma 4 give immediately the following result. COROLLARY 5. If R is a uniquely clean ring and e 2 = e ∈ R, then eRe is uniquely clean. COROLLARY 6. Every uniquely clean ring R is directly finite (ab = 1 implies ba = 1). Proof. If ab = 1, then ba is a (central) idempotent, so ba = ba(ab) = a(ba)b = 1. https://www.cambridge.org/core/terms. https://doi.