Ore Extensions Over Weakly 2-primal Rings And M-π-Armendariz Property

In recent years, Ore extensions have become an important research object in algebra. It mainly contains the following research directions. Firstly, whether the Ore extension of some class of rings is this class of rings; Secondly, it is studied for some properties and structures of Ore extension rings. This paper is researched on the first direction including Ore extensions of weakly semicommutative rings, weakly 2-primal rings, nil-semicommutative rings, nil-sc rings, Baer rings and quasi-Baer rings. Furthermore, in the condition of weakly 2-primal rings, we also study some properties of M-π-Armendariz rings. This thesis mainly consists of the following components:Chapter 1:We introduce the background, development process and research status of Ore extensions, and briefly summarize some important work and results in the literature.Chapter 2:We introduce some essential concepts, such as weakly semicommutative rings, nil-semicommutative rings, weakly 2-primal rings, (α,δ)-compatible rings, NI rings, a-skew Ar-mendariz rings, Baer rings, quasi-Baer rings and (α,δ)-skew Armendariz rings, and some results which were frequently used in the sequel.Chapter 3:We mainly investigate Ore extensions over weakly 2-primal rings. Let a be an endomorphism of a ring R and δ an a-derivation of R. The main results are the following:(1) If R is an (α,δ)-compatible weakly 2-primal ring, then R[x; α,δ] is a weakly semicommutative ring; (2) If R is an (α,δ)-compatible (α,δ)-skew Armendariz ring, then R is a nil-semicommutative ring if and only if R[χ; α, δ]is a nil-semicommutative ring; (3) If R is an (α,δ)-compatible ring, then R is a weakly 2-primal ring if and only if R[χ; α,δ] is a weakly 2-primal ring; (4) If R is a weakly (α,δ)-compatible ring and nil(R) is an ideal of R, then R is a weakly (α,δ)-skew Armendariz ring.Chapter 4:This chapter extends the notations of M-Armendariz rings and π-Armendariz rings, and investigates π-Armendariz rings relative to a monoid, referred to as M-π-Armendariz rings. We mainly discuss the relationships between M-π-Armendariz rings and related rings, and consider some extensions of M-π-Armendariz rings. Moreover, we also investigate the relationships between M-π-Armendariz rings and weak annihilator ideals.Chapter 5:We introduce some properties of skew polynomial rings and prove that:(1) If R is an a-Armendariz ring, then J(R[χ;α]) ∩ R is nil; (2) If R is an a-Armendariz ring, then R is an a-Baer ring if and only if R[x; a] is an a-Baer ring; (3) If R is an a-Armendariz ring and satisfies Ca condition, then R is an a-Baer ring(resp., a right a-p.q.-Baer ring, a right zip ring)if and only if R[χ; α] is an a-Baer ring(resp., right a-p.q.-Baer ring, a right zip).Chapter 6:Given a summary about Ore extensions of these classes of rings in this article, we roughly forecasted their possible applications and other research in the future.