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exponential ring

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Idea

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

Definitions

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

e 0=1e^{0} = 1
e a+b=e ae be^{a + b} = e^a \cdot e^b

Properties

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

Trigonometric functions

If an exponential ring RR has elements ii and 12\frac{1}{2} in RR such that ii=1i \cdot i = -1 and 12+12=1\frac{1}{2} + \frac{1}{2} = 1, then the sine and cosine could be defined as

sinx12i(e ixe ix)=12(e xie xi)i \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
cosx12(e ix+e ix) \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 a1e^a \coloneqq 1 for all aa in RR.

  • Any cyclic ring /2n\mathbb{Z}/2n\mathbb{Z} with e ()e^{(-)} defined such that e a=1e^{a} = 1 for a0mod2a \equiv 0 \mod 2 and e a=1e^{a} = -1 for a1mod2a \equiv 1 \mod 2 is an exponential ring.

  • The real numbers and complex numbers with the usual exponential map exp\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.