Definition (Ring)

A ring is a triple $(R, +, \cdot)$ where

• $(R,+)$ is an abelian group
• $(R,\cdot)$ is a semigroup
• Distributivity: $$\begin{align*} a\cdot(b+c)=ab+ac\\ (b+c)\cdot a=ba+ca \end{align*}$$

Definition (Unit)

The units in a ring $R$ are denoted by adding a star, $R^*$, symbolizing the invertible elements of the ring.