nLab exponential ring

Idea

Exponential rings are rings that behave like the real numbers with its exponential map.

Definitions

A ring $R$ is a exponential ring if it is equipped with a monoid homomorphism $e^{(-)}:R \to R$ 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

$$\sin{x} \coloneqq \frac{1}{2}\cdot i\cdot (e^{-i\cdot x} - e^{i\cdot x}) = \frac{1}{2}\cdot (e^{-x \cdot i} - e^{x \cdot i}) \cdot i$$

$$\cos{x} \coloneqq \frac{1}{2}\cdot (e^{i\cdot x} + e^{-i\cdot x})$$

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.

Related concepts

• exponential map

• Pythagorean ring

External links

• Wikipedia, Exponential ring