## Graduate Research Theses & Dissertations

#### Title

The classification of ideals

1965

#### Document Type

Dissertation/Thesis

Sexauer, Norman E.

#### Degree Name

M.S. (Master of Science)

#### Department

Department of Mathematics

Ideals (Algebra)

#### Abstract

The purpose of this paper is to initiate a study of the following problem: Given a ring, classify the ideals of this ring. In general this is a difficult problem. However, by taking specific rings it is possible to classify their ideals. This paper discusses rings and ideals in a brief but general manner. Then the ideals of the ring of integers, the ring of polynomials over a field, residue class rings, and matrix rings over an arbitrary ring with unity are determined. The ideals of the ring of integers and the ring of polynomials are classified in a similar manner, that is, all of the ideals of these two rings are principal ideals. Furthermore, prime and primary ideals provide us with a subclassification for the ideals of these rings. Prime integers and irreducible polynomials generate non-trivial prime ideals, while prime integers and irreducible polynomials raised to positive integral powers generate non-trivial primary ideals. Residue class rings have ideals whose elements belong to the set (b) ={ab|aEI/(m)}. Residue class rings have a finite number of elements, end hence, their ideals and the elements belonging to these ideals are also finite in number. Therefore, we are able to classify the ideals in these rings by listing them. In the ring of n by n matrices over an arbitrary ring with unity, we have ideals which are classified as follows. If R is a ring with unity, if Rn is the ring of n by n matrices over R, if J is a subset of R, and if Jn is the set of matrices over J, then the ideals of Rn are of the form Jn, where J is an ideal of R.

Includes bibliographical references (leaf [35])

[35] pages

eng

#### Publisher

Northern Illinois University