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Rings, shapes, and dimensions: A thread

(This is not meant to be too rigorous, so please bear with me)

So "rings" are sets "closed" (we'll see what this word means later) under addition, subtraction, and multiplication. That's three of the four basic arithmetic operations.

Kind of weird, compared to "groups" (one may think of them as being closed under two of the four operations - either addition/subtraction or multiplication/division) or "fields" (closed under all four). But that only makes rings more interesting, as we will see.

So "closed" here means that if you add, subtract, or multiply any two things in the set, you get another thing in the set.

For example, if you add, subtract, or multiply two integers, you get another integer. So the integers are a ring.

The integers are not closed under division! Sure, if you divide 6 by 2 you get 3, which is an integer...but if you divide 1 by 2, you get 1/2, which is not an integer.

On the other hand, the rational numbers are closed under division too, which makes them a field.

I should say, when we say "closed under division", we don't include 0.

Now rings not being closed under division only make them more interesting. Not every element can be divided by every other element.

So it's now interesting to see which ones can be divided by which!

This leads to notions like divisibility, primes, factorization, and so on.

Now, rings are often associated to shapes. The classic example is the functions (maps) from a shape that assign a number to a point. These functions form a ring.

So rings are not just numbers, functions can also form a ring. You can add, subtract, and multiply them.

Some functions of a special kind can be used to determine the shape exactly - when you know the functions, you know the shape.

Or, put another way, you know the ring, you know the shape.

For polynomial functions and shapes called "varieties" (shapes defined by polynomial equations, like a parabola, etc.) this follows from something called Hilbert's Nullstellensatz.

This is one motivation for the subject of algebraic geometry.

Now, one property of a shape is its dimension. A line or a curve has dimension 1. A plane or a surface has dimension 2, and so on.

Now, we said, if you know the ring, you know the shape (for some cases anyway). The dimension of a shape can also be determined from properties of the ring!

To do this, one must consider the "ideals" of the ring. These are special subsets of the ring "closed" (it's that word again) under addition among themselves and "scalar multiplication" by elements of the ring.

Ideals are kind of like vector spaces then, since we can add and scale their elements to form another. In fact, both ideals and vector spaces are examples of the more general concept of a "module".

The ideals that must be used to understand dimension are the prime ideals. They are so called because they generalize a property of prime numbers.

See, ideals were originally invented for number theory. But again, rings are not just numbers!

So to determine the dimension, we see if a prime ideal is contained in a prime ideal, and so on. They form a "chain" of prime ideals, and the length of this (if finite) is called the "Krull dimension" (this is the most sci-fi sounding term I've seen in math).

So if a ring has Krull dimension 1, perhaps it is the set of functions on a line or a curve! We haven't even seen the shape, but we can say something about it.

You know the ring, you know the shape.

Except there are some complications. See, the variety called an "elliptic curve" has Krull dimension 1, but if we put "complex" solutions to the equation that defines it it looks like a surface of a donut, not at all a curve. "Real" solutions still work fine though.

These things are not too bad, since we can still make them consistent. But it is true that dimension is still complicated! That is why there is an entire subject called "dimension theory", to see if the different notions of dimension all make sense, and when.

So, I'll end this thread by saying I probably like dimension 1 most. My favorite dimension 1 shapes are the elliptic curves, and...the integers.

They are a ring after all, and determine a shape.

You know the ring, you know the shape. At least we should...

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