Polynomial Extensions And Generalized Power Series Rings And Modules

This dissertation, consisting of seven chapters, is concerned with the problem of polynomial extension of rings and the properties of generalized power series rings and modules. Unless otherwise mentioned, in this thesis,R will denote an associative ring with identity, andαan endomorphism,δanα-derivation of a ring R.Chapter 1 introduces the background and the main results obtained in this thesis.Chapter 2 introduces weak symmetric rings and weak (α,δ) -symmetric rings, which are generalizations of symmetric rings, and studies weither the weak symmetric or weak (α,δ)- symmetric properties of rings can be preserved in their polynomial extension rings. In this chapter, we first investigate the properties of weak symmetric rings, and show that all symmetric rings are weak symmetric, and the upper triangular matrix ring over R is a weak symmetric ring if and only if R is a weak symmetric ring; Secondly, we study the relationship between the ring R and the Ore extension R[x;α,δ],and prove that R[x;α,δ] is a weak symmetric ring if and only if R is a weak symmetric ring when R is an ((?),(?))-compatible and reversible ring. Finally, we see that if R is a semicommutative ring, then R is a weak (α,δ)-symmetric ring if and only if R[x] is a weak ((?),(?))- symmetric ring, where (?) and (?) is the extended map ofαandδ,respectively.Chapter 3 defines and studies weak (α,δ)-compatible rings and weak (α,δ)-Armendariz rings, which are generalizations ofα-rigid rings. We mainly obtain in this chapter that if R is a weak (α,δ)-compatible and semicommutative ring, then R is a weak (α,δ)-Armendariz ring, and if R is a weak (α,δ)-compatible ring, and R[x] a semicommutative ring, then R[x] is a weak (α,δ)-Armendariz ring where (?) and (?) is the extended map ofαandδ,respectively.Chapter 4 investigates weak a-rigid rings and weakα- skew Armendariz rings, which are generalizations ofα-rigid rings andα- skew Armendariz rings, respectively. In this chapter, we first discuss the properties of weakα-rigid rings and show that a weakα-rigid ring is a true generalization of anα- rigid ring by providing some examples. Then we investigate the relationship between weakα- rigids and weakα- skew Armendariz rings, and see that if nil(R) is an ideal of a ring R,then a weakα-rigid ring is a weakα- skew Armendariz ring, and if R is a weakα-rigid semicommutative ring, then R[x] is a weakα-skew Armendariz ring.Chapter 5 is concerned withα-McCoy rings and weak McCoy rings, which are natural generalizations of McCoy rings, and studies weither the weak McCoy properties of a ring R can be preserved in the Ore extension R[x;α,δ].In this chapter, we first obtain that anα- McCoy ring is a true generalization of a McCoy ring by compairingα- McCoy rings with McCoy rings, and show that if R is anα-compatible and reversible ring, then R is anα-McCoy ring. Thus P.P. Nielson's Theorem 2 in [76] is a direct corollary of our result. Finally, we obtain that if R is a reversible and (α,δ)-compatible ring, then the weak McCoy properties of a ring can be preserved in the Ore extension R[x;α,δ]. Consequently, several known results on McCoy rings are extended to a general setting.Chapter 6 discusses weak Zip rings and studies weither the weak Zip properties of rings can be preserved in the polynomial extension rings. As a generalization of annihilator,in this chapter we introduce weak annihilator and investigate their properties. Then based on the properties of weak annihilator, we define weak Zip rings and obtain that the weak Zip properties of a ring R is preserved in the Ore extension R\x;α,δ] when R is an (α,δ)- compatible reversible ring.Chapter 7 studies the properties of generalized power series rings and generalized power series modules. In Section 7.2, we investigate the GM-property of formal triangular matrix rings, and prove that the GM-property of formal triangular matrix ring over a ring R is preserved in the formal triangular matrix ring over a generalized power series ring. In Section 7.3, we obtain that the K₀-group of [[R^S,≤]] is isomorphic to the K₀-group of R. Thus the K₀-groups of some rings of generalized power series are explicitly described. Moreover, we study the stable range properties of the Morita Context over generalized power series rings. We show that if rings A and B are (s,2)-rings, then so is the Morita Context ([[A^S,≤]],[[B^S,≤]],[[M^S,≤]],[[N^S,≤]],ψ^S,φ^S) over generalized power series rings. Also we get analogous results for unit 1-stable range rings and rings which have stable range one. These give new classes of rings satisfying such stable range conditions. In Section 7.4, we investigate the Armendariz properties of modules over generalized power series rings. As a generalization of Armendariz rings, we introduce S-Armendariz modules over generalized power series rings, and prove that the S-Armendariz modules over generalized power series rings have many properties similar to Armendariz rings. As a application of S-Armendariz modules, we prove that if R is a ring, and M an S - Armendariz module, and for eachφ² =φ∈[[R^S,≤]], there exists e² = e∈R such thatφ= C_e,then M_R is a Baer module, quasi-Baer module or p.p. module if and only if [[M^S,≤]] is a Baer module, quasi-Baer module or p.p. module, respectively.