A ring is clean (resp. almost clean) if each of its elements is the sum of a
unit (resp. regular element) and an idempotent. In this paper we define the
analogous notion for *-rings: a *-ring is *-clean (resp. almost *-clean) if its
every element is the sum of a unit (resp. regular element) and a projection.
Although *-clean is a stronger notion than clean, for some *-rings we
demonstrate that it is more natural to use.

The theorem on cleanness of unit-regular rings from [V. P. Camillo, D.
Khurana, A Characterization of Unit Regular Rings, Communications in Algebra,
29 (5) (2001) 2293-2295] is modified for *-cleanness of *-regular rings that
are abelian (or reduced or Armendariz). Using this result, it is shown that all
finite, type I Baer *-rings that satisfy certain axioms (considered in [S. K.
Berberian, Baer *-rings, Die Grundlehren der mathematischen Wissenschaften 195,
Springer-Verlag, Berlin-Heidelberg-New York, 1972] and [L. Vas, Dimension and
Torsion Theories for a Class of Baer *-Rings, Journal of Algebra, 289 (2)
(2005) 614-639]) are almost *-clean. In particular, we obtain that all finite
type I AW*-algebras (thus all finite type I von Neumann algebras as well) are
almost *-clean. We also prove that for a Baer *-ring satisfying the same
axioms, the following properties are equivalent: regular, unit-regular, left
(right) morphic and left (right) quasi-morphic. If such a ring is finite and
type I, it is *-clean. Finally, we present some examples related to group von
Neumann algebras and list some open problems.