Rings with Central Idempotent or Nilpotent Elements

Nilpotent Elements in Skew Polynomial Rings

 Letbe a ring with an endomorphism and an -derivationAntoine studied the structure of the set of nilpotent elements in Armendariz rings and introduced nil-Armendariz rings. In this paper we introduce and investigate the notion of nil--compatible rings. The class of nil--compatible rings are extended through various ring extensions and many classes of nil--compatible rings are constructed. We al...

Rings in which elements are the sum of an‎ ‎idempotent and a regular element

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...

P -nilpotent Completion Is Not Idempotent

Let P be an arbitrary set of primes. The P -nilpotent completion of a group G is defined by the group homomorphism η : G → G P̂ where G P̂ = invlim(G/ΓiG)P . Here Γ2G is the commutator subgroup [G,G] and ΓiG the subgroup [G,Γi−1G] when i > 2. In this paper, we prove that P -nilpotent completion of an infinitely generated free group F does not induce an isomorphism on the first homology group with...

Central Units of Integral Group Rings of Nilpotent Groups

In this paper a finite set of generators is given for a subgroup of finite index in the group of central units of the integral group ring of a finitely generated nilpotent group. In this paper we construct explicitly a finite set of generators for a subgroup of finite index in the centre Z(U(ZG)) of the unit group U(ZG) of the integral group ring ZG of a finitely generated nilpotent group G. Ri...

Extensions of Nilpotent P.P. Rings