Different kinds of rings
There are a number of different kinds of rings differentiated (sorry) by properties of their multiplications. Most of them lead into their own specialized areas of study. I mentioned that a ring may or may not be commutative, and it may or may not have an identity, but there are a few more that will be useful.
One initially counterintuitive idea is that it’s entirely possible that a ring has “zero divisors”: two nonzero elements that multiply to give zero. Imagine starting with two copies of the integers, and , writing elements of the second copy as integers with a bar over them. Now consider pairs of elements, one from each copy, . Add pairs by adding the two components, but multiply them like this:
Notice that the product of any two elements of is zero! Weird. Eerie.
To be explicit: an element of this ring coming from is . We calculate the product:
So, any element for which there is a so that is called a left zero divisor. Right zero divisors are defined similarly. If a ring has no zero divisors, so the product of two nonzero elements is always nonzero, we call it an “integral domain”. The integers are just such an integral domain, fittingly enough.
Now if a ring has a multiplicative identity we can start talking about multiplicative inverses. We say an element has a left inverse if , or a right inverse if . If a ring has both a left and a right inverse they’re the same, since
In this case we call a unit and write its inverse as . We can also see that an element having a left (right) inverse cannot be a left (right) zero divisor:
If every nonzero element of a ring is a unit, we call it a division ring.
In the case of commutative rings, all these distinctions between “left” and “right” (zero divisors, inverses, etc.) disappear, since multiplication doesn’t care about the order of the factors. We actually have a special name for a commutative division ring: we call it a “field”, though everyone else in the world except the Belgians seems to call it a “(dead) body” (körper, corps, поле, test, lichaam, …).
[EDIT: added explicit calculation verifying that elements from in the example are zero-divisors.]