Exponentiation in Rings

Abstract

1. EXPONENTLSL RINGS. Adding an element of a group to itself may be viewed as multiplying the element by two. Analogously, multiplying an element of a ring by itself may be viewed as raising it to the second power. Although the theory of groups with a multiplication (i.e., the theory of rings) is very well developed, there seems to be no axiomatized theory of rings with an exponentiation. The purpose of this paper is to suggest a formal setting for such a theory and to determine some of the properties of the resulting structures. Underlying the exposition are the general references [1, 2]. The natural numbers may be used as exponents for all the elements of any commutative ring. For the real numbers, there are more possibilities. For instance, re exists for all real numbers e and all positive real numbers r. With this in mind, we consider bases and exponents separately for the general case. Specifically, let R be a ring, let B (the bases) be a multiplicative subsemigroup of (R, * ) which does not contain 0, and let E (the exponents) be a semiring with unit element 1. (That is, (E, +) is a semigroup, (E, * ) is a semigroup with 1, and distributes over + from both the left and the right.) A binary operation B X E B c R ((b, e) be)