ABSTRACT
This chapter focuses on groups in abstract algebra. Groups possess a single binary operation, to classes of sets whose algebraic structure involves two binary operations. The first class that the chapter focuses is that of objects called rings. The two operations of a ring R are thought of as addition and multiplication. Under addition, the non-empty set R must form an Abelian group, but this is not necessarily true of multiplication in R. In a ring, there is no requirement that under multiplication every element have a multiplication inverse, so there is no need to throw out elements in order to satisfy that the set is a ring (as long as the eight necessary properties are satisfied, of course). One way to describe characteristic of a ring is that it is the order of the additive cyclic group <1> generated by the unity element 1 if that group is finite, and it is 0 if that group is infinite.
