symmetric monoidal (∞,1)-category of spectra
Exponential rings are rings that behave like the real numbers with its exponential map.
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 $R$:
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.
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.
Every ring can be made into a exponential ring by defining $e^a \coloneqq 1$ for all $a$ in $R$.
Any cyclic ring $\mathbb{Z}/2n\mathbb{Z}$ 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.
Last revised on June 5, 2021 at 13:19:33. See the history of this page for a list of all contributions to it.