Rings with infinite sums

In some sense, ring theory can be considered a formalization of the operations of addition and multiplication. The operation that we study in this thesis appears to be the process of forming an infinite sum. It is probably one of the basic nontrivial operations of beginning college mathematics.; In this thesis, we first present an abstract definition of a ring with infinite sum, then concentrate on the more interesting case of a graded ring with infinite sum, which we will call the {dollar}g{dollar}-sum. We study the relationship between a graded ring and its {dollar}g{dollar}-sum extension and prove the existence of nontrivial {dollar}g{dollar}-sums and maximal {dollar}g{dollar}-sums over a graded ring. We then study special cases in matrix algebra and {dollar}Z{dollar}-graded {dollar}g{dollar}-sums. Finally, we present generalizations of several classic ring structure theorems to the {dollar}g{dollar}-sums.