Armendariz Quaternion Algebras And Armendariz Generalized Trivial Extensions

Armendariz rings is introduced by Rege and Chhawchharia in 1997. Since then Armendariz rings have attracted attentions of many researchers. Many papers have be published on this topic in the recent years.A ring R is called an Armendariz ring if it satisfies the following condition:This kind of rings is named after Armendariz because Armendariz proved in 1974 that reduced rings (the rings without nonzero nilpotent elements) have the property above.It is well known that subrings of an Armendariz ring are Armendariz, but a quotient ring of an Armendariz ring need not be Armendariz. So the following questions arises naturally:1. Which kind of extensions of an Armendariz ring are Armendariz?2. Which quotient rings of an Armendariz ring are Armendariz?In this dissertation, we are concerned with the Armendariz property on the gen-eralized quaternion algebra Q over a field, Q[x]/(x2+1), and two kinds of generalized trivial extension of Armendariz rings.This dissertation is organised as follows.In Section 2, we discuss the Armendariz property on the generalized quaternion algebra Q=(α,β/F) over a field F and Q[x]/(x2+1). We first present necessary and sufficient conditions for generalized quaternion algebras to be Armendariz. It is proved that Q is Armendariz if and only if Q is reduced. Then we consider the Armendariz property of Q[x]/(x2+1). Finally the following result are proved.Theorem 2.12. Let Q be a real quaternion algebra, then Q[x]/(x2+1) is not an Abel ring, and so it is not Armendariz.Theorem 2.13. H[x]/(x2+q) with q ∈ H is an Armendariz ring if and only if q is not a positive real number.In Section 3, we focus attention to the Armendariz property of two kinds of generalized trivial extensions.Let h be an endomorphism of a ring R, M be an (R,.R)-bimodule. Defind multiplication on the direct product T= R×M of additive groups R,M as follows:Then T is a ring with 1, denoted by R ∝h, M. If h is the identity map, then T is called a trivial extension of R, denoted by R. ∝ M.Theorem 3.6. Let h:R �' R be a ring homomorphism and let M be an (R,Rh)-bimodule. Then T= R ∝h M is an Armendariz ring if and only if the following conditions hold:1. R is an Armendariz ring;2. M is an Armendariz (R,Rh)-bimodule;3. For f(x),g(x) ∈ R[x] with f(x)g(x)= 0, we have f(x)M[x] ∩ M[x]gh(x)= 0.Theorem 3.10. Let M,N be both (R,Rh)-bimodules. Then the following conditions are equivalent.1. T(R, M, N) is an Armendariz ring.2. (R ∝ M) ∝ N is an Armendariz ring.3. R ∝ (M × N)is an Armendariz ring.4. R ∝ M and R ∝ N are both Armendariz rings.5. The following conditions hold:(a) R is an Armendariz ring;(b) M, N are both Armendariz (R,R)-bimodules;(c) If f(x)g(x)= 0 for f(x),g(x) ∈ R[x],then...