Nicholas Bastian; Jaden Brewer, Southern Utah University
Schur rings are a type of algebra that is spanned by a partition of finite groups that meets other conditions. Schur rings were originally developed by Schur and Wielandt in the first half of the 20th century. They were originally developed to study permutation groups and have since been more widely studied, especially in the 1980s and 1990s, to look at finite cyclic groups, which are a finite sets that cycle through its elements equipped with an operation satisfying certain properties. Past research has provided a classification of Schur rings over finite cyclic groups. We will provide an extension to of this classification to Schur rings over infinite cyclic groups. This will be accomplished by using mapping techniques that proved successful when considering Schur rings over finite cyclic groups. Using these mapping techniques we will argue that any arbitrary Schur ring over the integers must in fact be essentially the same as one of the two Schur rings we know to exist over the integers.