let r be an associative ring with unity. an element a in r is said to be r-clean if a = e+r, where e is an idempotent and r is a regular (von neumann) element in r. if every element of r is r-clean, then r is called an r-clean ring. in this paper, we prove that the concepts of clean ring and r-clean ring are equivalent for abelian rings. further we prove that if 0 and 1 are the only idempotents...

let r be an associative ring with unity. an element a in r is said to be r-clean if a = e+r, where e is an idempotent and r is a regular (von neumann) element in r. if every element of r is r-clean, then r is called an r-clean ring. in this paper, we prove that the concepts of clean ring and r-clean ring are equivalent for abelian rings. further we prove that if 0 and 1 are the only idempotents...

We organize this set of notes around a few theorems of Weierstrass. Write OCn for the sheaf of holomorphic functions on C. In the first three sections, we deduce the following “algebraic” results as consequences of the Weierstrass theorems: (i) for each open set Ω ⊂ C, the ring OCn(Ω) is not Noetherian, (ii) the local ring OCn,0 is factorial, (iii) the local ring OCn,0 is Noetherian. The two la...