In this paper, we study number rings and their factorization properties. We begin with an introduction to number fields and number rings. Then we consider the ring Z[i], which is a unique factorization domain, where each element factors uniquely into a product of prime elements. We classify all irreducible elements in this ring. Then we consider an example of non-unique factorization in the ring Z[sqrt(-5)], which is not a unique factorization domain. Then we introduce the idea of a Dedekind domain. The main discussion of the paper is to prove that number rings are Dedekind domains, then to prove that in a Dedekind domain, every ideal factors uniquely into a product of prime ideals. Then we consider an example of a ring that is not a Dedekind domain, and we find an example of non-unique prime ideal factorization. Finally, we conclude by proving how the primes split in the quadratic fields.