A study of δ-Quasi-Baer ring

ENGLISH :

Items that is studied at abstraction algebra basically about set and its operation, and always equivalent with a nonempty set which have elements are operated with one or more binary operation. A set is provided with one or more binary operation is called algebra structure. Algebra structure with one binary operation which fulfill the certain natures is called group. While for nonempty set with two binary operations which fulfill the certain natures is called ring. At other algebra structure is also studied about Quasi-Baer ring. Let R is Quasi-Baer ring if the right annihilator of every ideal (as a right ideal) is generated by idempotent.

Continuing from Quasi-Baer ring can be developed to become some discussions, those are δ-Quasi-Baer ring. Let δ be a derivation on R. A ring R is called δ-Quasi-Baer ring if the right annihilator rR (X) = {c∈R} ⃒dc = 0, ∀d∈X} of every δ-ideal of R is generated by idempotent. Let δ : R → R is derivation of R, that is, δ is an additive map such that, so that the extension of δ : R → R is δ ̅ (αb) = δ(α)b - δα(b),∀α,b∈R. For a ring R with a derivation δ, there exists a derivation on S = R[x; δ] which extends δ. Considering an inner derivation δ on S by x defined by δ ̅ (f(x))= xf(x) - f(x)x,∀f(x)∈S .Then, δ ̅ (f(x))= δ(α_0)+...+ δ(α_n)x^n for all f(x)=α_0+...+α_n x^n∈S and δ ̅ (r)= δ(r), ∀r∈R which means that δ ̅ is an extension of δ. We call such a derivation δ ̅ on S an extended derivation of δ. For each α∈R and nonnegative integer n, there exist t_0 ,..., t_n ∈Z such that x^n α = ∑_(i=0)^n t_i δ^n=i(α)x^i. Let R[x; δ] is the polynomial ring whose elements are the polynomials denote ∑_(i=0)^n r_i x^i ∈R, where the multiplication operation by xb = bx + δ(b),∀b ∈R. Three commonly used operations for polynomials are addition “ + “, multiplication “ . ” and composition “ ◦ “. Observe that (R[x], +, .) is a ring and (R[x], +, ◦) is a left nearring where the substitution indicates substitution of f(x) into g(x), explicitly f(x) .. g(x) = f(g(x) for each f(x), g(x), explicitly f(x)◦g(x) = f(g(x) for each f(x), g(x) ∈ R[x]. At other researcher is suggested to perform a research morely about δ- Quasi-Baer ring, with searching natures of others.