A ring R is called clean if every element of the sum a unit and an idempotent. Motivated by question proposed Lam on cleanness von Neumann Algebras, Vaš introduced more natural concept for ?-rings, ?-cleanness. More precisely, ?-ring ?- projection (?-invariant idempotent). Let F be finite field G abelian group. In this paper, we introduce two classes involutions group rings form characterize ?-...

in this paper, we introduce the new notion of n-f-clean rings as a generalization of f-clean rings. next, we investigate some properties of such rings. we prove that mn(r) is n-f-clean for any n-f-clean ring r. we also, get a condition under which the de nitions of n-cleanness and n-f-cleanness are equivalent.

We prove the cleanness of cuspidal character sheaves in arbitrary characteristic in the few cases where it was previously unknown. 2010 Math. Subj. Class. 20G99.

BACKGROUND Patient's satisfaction with both private and public laboratory services is important for the improvement of the health care delivery in any country. METHODS A cross-sectional survey was conducted in 24 randomly selected health facilities with laboratories that are conducting HIV related testing, in Mainland Tanzania. The study assessed patient's satisfaction with the laboratory ser...