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...

for a ring endomorphism $alpha$ and an $alpha$-derivation $delta$, we introduce a concept, so called skew $pi$-armendariz ring, that is a generalization of both $pi$-armendariz rings, and $(alpha,delta)$-compatible skew armendariz rings. we first observe the basic properties of skew $pi$-armendariz rings, and extend the class of skew $pi$-armendariz rings through various ring extensions. we nex...

In this paper we construct a free resolution for a free partially commutative monoid and with its help prove the Husainov’s Conjecture. We follow the ideas of D. Cohen who built in [3] a resolution for the so-called graph product of groups, given resolutions for factors. The presentation of the graph product with the help of direct and free amalgamated products played the leading role at that. ...

When studying Haken 3-manifolds, one is led naturally to the following construction: the amalgamation of two 3-manifolds M and M ′ via a homeomorphism between their boundaries. In this paper, we study the behaviour of Heegaard genus under this operation. We show that, provided the gluing homeomorphism is ‘sufficiently complicated’ and M and M ′ satisfy some standard conditions, then the Heegaar...

The problem of integrating knowledge from multiple and heterogeneous sources is a fundamental issue in current information systems. In order to cope with this problem, the concept of mediator has been introduced as a software component providing intermediate services, linking data resources and application programs, and making transparent the heterogeneity of the underlying systems. In designin...

We show that the real Cremona group of plane is a non-trivial amalgam two groups amalgamated along their intersection and give an alternative proof its abelianisation.

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.