Wheels – on division by zero

Abstract

We show how to extend any commutative ring (or semiring) so that division by any element, including 0, is, in a sense, possible. The resulting structure is called a wheel. Wheels are similar to rings, but $0x=0$ does not hold in general; the subset $\{x|0x=0\}$ of any wheel is a commutative ring (or semiring), and any commutative ring (or semiring) with identity can be described as such a subset of a wheel. The main goal of this paper is to show that the given axioms for wheels are natural, and to clarify how valid identities for wheels relate to valid identities for commutative rings and semirings.