The Cantor set is a fractal that is obtained by repeatedly removing the middle third of a segment. Start with the closed interval . Remove the open interval to obtain , i.e. two disjoint closed segments. Remove the middle thirds of those two segments, and you end up with four disjoint segments. After infinitely many steps, the result is called the Cantor set. The diagram below is only an approximation after a finite number of steps.
The point of this post is to provide examples and non-examples for the following ring classes:
- Commutative Ring
- Integral Domain
- Integrally Closed Domain
- Unique Factorization Domain (UFD)
- Principal Ideal Domain (PID)
- Euclidean Domain
Given that the name of this site is Epic Math, I felt I should write about a topic where the word epic is a technical term! It is hardly surprising that such a term exists, as many otherwise non-mathematical words such as almost, simple, open, connected, regular, normal, field, ring, onto, map, twin, lucky, and even sexy have technical definitions.
The term epic is found in category theory, which is an extremely abstract branch of mathematics that formally deals with many other fields of math. It is so strange that some mathematicians have labeled it “abstract nonsense.”
Sequences of 0’s and 1’s
Suppose I want to write a sequence of 0’s and 1’s that contains every possible 2-letter subsequence. This means somewhere in my sequence I need 01, 10, 00, and 11.
Obviously, gluing them all together gives a valid sequence: