Graded Extensions In A Skew Laurent Power Series Ring

As an important class of rings, non-commutative valuation rings are of great significance in the study of non-commutative ring theory. At the end of the last century, with the question of non-commutative valuation ring extensions was put forward by H.H.Brungs, G.Torner and G.Schroder, the study of non-commutative valuation rings have achieved great progress. As a special non-commutative valuation ring extension, graded extension also have great value to study.Graded extensions in K((X, σ)) are discussed in this thesis. Let V be a total valuation ring of a division ring K with an automorphism a and let K((X, σ)) be a skew Laurent power series ring of K.Suppose that is a graded extension of V in K((X, σ)). We can divide it into two cases:non-trivial graded extensions and trivial graded extensions.1. Suppose that A is a non-trivial graded extension. Let W= Ol(A1), which is an over-ring of V. There are two cases:A1 is a finitely generated left W-ideal or A1 is not a finitely generated left W-ideal.Case (1) Suppose that A1 is a finitely generated left W-ideal. We can divide it into five cases: Type (a) W= V,A1= Vα= ασ[V), A-1= Vσ-1(α-1); Type (b) A1= Wα(?) ασ(W); Type (c) A1= Wα (?) ασ(W)(W (?) V) Type (d) A1= Wα= ασ(W), A-1= J(W)σ-1(α-1), J(W)2 = J (W)(J(W) is the Jacobson radical of W); Type (e) A1= Wα= ασ(W), A-1= J(W)σ-1(α-1), J(W)= Wb-1(b ∈K).Case (2) Suppose that A1 is not a finitely generated left W-ideal. We can divide it to three cases: Type (f) * A1 (?) A1; Type (g) * A1 = A1,* Mi is not a principal left W-ideal for any i ∈N; Type (h)*A1= A1;*M1 is a principal left W-ideal for some l ∈ N.2. Suppose that A is a trivial graded extension. We have the follow result:There is a kind of graded extensions in K((x,a)) which is similar to commutative case. It is invariant graded extensions. We study invariant graded extensions in §2.3 and give a necessary and sufficient condition for a graded extension to be an invariant graded extension.This thesis is composed of two parts. The first part is the introduction. The second part is the main body of this thesis, which includes chapters 1-2.In part 1, some of the research background、some definitions and properties are in-troduced. In Chapter 1, non-trivial graded extensions、trivial graded extensions、invariant graded extensions are studied and some examples of non-trivial graded extensions、trivial graded extensions studied are given. In Chapter 2, some properties of graded extensions in a skew Laurent power series ring are studied.