Ziembowski, Michal

2018-01-31T11:40:59Z

2018-01-31T11:40:59Z

2010

Prüfer domains are commutative domains in which every non -zero finitely generated ideal is invertible. Since such domains play a central role in multiplicative ideal theory, any equivalent condition to the Prüfer domain notion is of great interest. It occurs that the class of Prüfer domains is equivalent to other classes which are investigated in theory of commutative rings (see [22D. Namely, for commutative rings the following classes are equivalent:

(1) Semihereditary domains.

(2) Domains which have weak dimension less or equal to one.

(3) Distributive domains.

(4) Gaussian domains.

(5) Prüfer domains.

Many authors have studied so called Prüfer rings which are a generalization of notion of Prüfer domains to the case of commutative rings with zero divisors. In this context there are investigated the following classes of commutative rings:

(I) Semihereditary rings.

(II) Rings which have weak dimension less or equal to one.

(III) Distributive rings.

(IV) Gaussian rings.

(V) Prüfer rings.

Recently the main stress in the area is focused on Gaussian rings (e.g. see [8] or [22]). In [22] S. Glaz showed that we have (I) —» (II) —» (III) —» (IV) —» (V) and no one of these implications can be replaced by the equivalence.

In this thesis the notion of a Gaussian ring is extended to the noncommutative setting by introducing a new class of rings which are called right Gaussian rings. We investigate the relations with noncommutative analogs of classes (I), (II), (III), (IV), and in some cases (V). Moreover, we study some related subjects which naturally occur during our research concerning right Gaussian rings.

In Chapter 2 we recall some facts regarding right distributive rings, and define right Gaussian rings. Moreover, we study basic properties of right Gaussian rings. We also present results about the connection between the above classes of rings.

Chapter 3 includes an investigation about right Gaussian skew power series rings. We will give an extension to the noncommutative case of a well -known result by Anderson and Camillo (see [2, Theorem 17]).

In Chapter 4 we define skew generalized power series rings and for positively ordered monoids we describe those of above which are right Gaussian.

It occurs that for a right Gaussian ring a ring of quotients may not exist, and even when it exists, it need not be right Gaussian. We study relevant these issues formulate Chapter 5.

In Chapter 6 we consider a class of homomorphie images of a polynomial ring R[x] and give the necessary and sufficient conditions for a ring R under which these images are right Gaussian.

In Chapter 7 we make an effort to establish what kind of relations hold among right Gaussian rings, right Prüfer rings and some other classes of noncommutative rings.

Right Gaussian rings are exactly right duo Armendariz rings. This fact is a reason to take on Armendariz rings in detail, which we do in Chapter 8.

The final chapter contains investigations about some subclasses of unique product monoids which appear naturally in Chapter 8.

http://hdl.handle.net/1842/27743

The University of Edinburgh

Annexe Thesis Digitisation Project 2017 Block 16

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Right Gaussian rings and related topics

Thesis or Dissertation

PhD Doctor of Philosophy