A ring$R$ is a exponential ring if it is equipped with a monoidhomomorphism$e^{(-)}:R \to R$ from the additive monoid of $R$ to the multiplicative monoid of $R$:

$e^{0} = 1$

$e^{a + b} = e^a \cdot e^b$

Properties

The monoid homomorphism is an group homomorphism into the multiplicative subgroup of two-sided units of $R$, due to the fact that the additive monoid of $R$ is a group.

Trigonometric functions

If an exponential ring $R$ has elements $i$ and $\frac{1}{2}$ in $R$ such that $i \cdot i = -1$ and $\frac{1}{2} + \frac{1}{2} = 1$, then the sine and cosine could be defined as

It could be proven from these definitions that the sine and cosine satisfy various trigonometric identities.

Examples

Every ring can be made into a exponential ring by defining $e^a \coloneqq 1$ for all $a$ in $R$.

Given a natural number $n$, the integers modulo $2n$$\mathbb{Z}/2n\mathbb{Z}$ with $e^{(-)}$ defined such that $e^{a} = 1$ for $a \equiv 0 \mod 2$ and $e^{a} = -1$ for $a \equiv 1 \mod 2$ is an exponential ring.

The real numbers and complex numbers with the usual exponential map$\exp$ are a exponential ring.

An exponential ring that is a field is a exponential field.