Okay, I know I’ve been doing a lot more high-level stuff this week because of the thing, but it’s getting about time to break some new ground.
A ring is another very well-known kind of mathematical structure, and we’re going to build it from parts we already know about. First we start with an abelian group, writing this group operation as . Of course that means we have an identity element , and inverses (negatives).
To this base we’re going to add a semigroup structure. That is, we can also “multiply” elements of the ring by using the semigroup structure, and I’ll write this as we usually write multiplication in algebra. Often the semigroup will actually be a monoid — there will be an identity element . We call this a “ring with unit” or a “unital ring”. Some authors only ever use rings with units, and there are good cases to be made on each side.
Of course, it’s one thing to just have these two structures floating around. It’s another thing entirely to make them interact. So I’ll add one more rule to make them play nicely together:
This is the familiar distributive law from high school algebra.
Notice that I’m not assuming the multiplication in a ring to be invertible. In fact, a lot of interesting structure comes from elements that have no multiplicative inverse. I’m also not assuming that the multiplication is commutative. If it is, we say the ring is commutative.
The fundamental example of a ring is the integers . I’ll soon show its ring structure in my thread of posts directly about them. Actually, the integers have a lot of special properties we’ll talk about in more detail. The whole area of number theory basically grew out of studying this ring, and much of ring theory is an attempt to generalize those properties.