## Student Work

7-1-1978

Thesis

#### Degree Name

Master of Arts (MA)

Mathematics

Dr. John Karlof

#### Abstract

Take G to be any multiplicative group. Let [G] = and choose q to be a prime such that n and q are relatively prime. Let K denote the field of order q (i.e. GF(q)-K). We form the group algebra KG defined to be the set of all formal sums {mathematical formula} with multiplication and addition defined by {series of formulas}. A straightforward application of these definitions yields that KG is an associative algebra with multiplicative identity. In fact, the identity in the group G acts as the multiplicative identity in KG. Definition 1.1.1. A ring is said to satisfy the minimum chain condition if it satisfies-the following two properties: {more formulas} The dimension of KG over Kasa vector space is n, and every ideal of KG is a vector subspace. ·Therefore, KG satisfies the minimum chain condition. {formula} s. the set of all products of k elements in I). The radical of the ring, (denoted Rad(R)), is the sum of all nilpotent left ideals.