Classification of Finite Rings of Order p 2

Most treatments of elementary abstract algebra include a discussion of finite groups and some work on their classification. However, very little is done with finite rings. For example most beginning texts state and prove the theorem that for p a prime the cyclic group of order p is the only group of order p up to isomorphism. Yet the equally striking and easily proved result that for a prime p there are only two rings of order p up to isomorphism is either not mentioned at all or relegated to the exercises. The purpose of this note is to give a complete classification of all finite rings of order p2 with p a prime. In particular, we show that up to isomorphism there are exactly 11 rings of order p2. The techniques are elementary and grew out of a project given to an undergraduate abstract algebra course. We use a concept called a ring presentation, which is an excellent computational tool for dealing with finite rings. After explaining this concept, we state our main result, Theorem 2 , which can be given as a large project to a good undergraduate class with guidance provided by the instructor.