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A ring R is said to be clean if every element of R is a sum of an idempotent and a unit. The class of clean rings is quite large and includes, for instance, semiperfect rings (and thus finite rings), and rings of linear transformations of vector spaces. We prove that the endomorphism ring of every continuous (or discrete) module is clean.

An element in a ring R is said to be clean (respectively unit-regular) if it is the sum (respectively product) of an idempotent element and an invertible element. If all elements in R are unitregular, it is known that all elements in R are clean. In this note, we show that a single unit-regular element in a ring need not be clean. More generally, a criterion is given for a matrix a b 0 0 to be clean in a matrix ring M 2 (K) over any commutative ring K. For K = Z, this criterion shows, for instance, that the unit-regular matrix 12 5 0 0 is not clean. Also, this turns out to be the "smallest" such example.

A classical result of Zelinsky states that every linear transformation on a vector space V, except when V is one-dimensional over ℤ2, is a sum of two invertible linear transformations. We extend this result to any right self-injective ring R by proving that every element of R is a sum of two units if no factor ring of R is isomorphic to ℤ2.

In this paper, we study the class of rings that satisfy internal direct sum cancellation with respect to their 1-sided ideals. These are known to be precisely the rings in which regular elements are unitregular. Further characterizations for such "IC rings" are given, in terms of suitable versions of stable range conditions, and unique generator properties of idempotent generated right ideals. This approach leads to a uniform treatment of many of the known characterizations for an exchange ring to have stable range 1. Rings whose matrix rings are IC turn out to be precisely those rings whose finitely generated projective modules satisfy cancellation. We also offer a couple of "hidden" characterizations of unit-regular elements in rings that shed some new light on the relation between similarity and pseudo-similarity-in monoids as well as in rings. The paper concludes with a treatment of ideals for which idempotents lift modulo all 1-sided subideals. An appendix by R.G. Swan 1 on the failure of cancellation for finitely generated projective modules over complex group algebras shows that such algebras are in general not IC.