The last post I made had a little terminology mix-up, but I liked reading the responses nevertheless.
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Rings you can’t necessarily divide by nonzero elements; fields you can.
One ring that’s “not numbers” is the ring of polynomials (with real coefficients) in one variable X. You can add, subtract, and multiply them — we teach middle-schoolers to do it — but you can’t divide polynomials and get a polynomial result. Thus, this isn’t a field.
However, what if we consider two polynomials to be “the same” if they take the same value at X=0. That is, if they have the same constant term. This is a quotient ring of the ring of polynomials, and it consists of equivalence classes of polynomials under this relation. Of course, it just happens to be isomorphic to the real numbers, so maybe you don’t like that example.
This is a tough one, I feel like the definitions of rings and fields are designed so that the rules, how one would calculate with whole numbers and reals, can be applied respectively.
The best I can do is maybe give you more tangible analogs of whole numbers and reals, that capture the spirit.
To visualize a ring, imagine a flat plane. You start of at some marked spot ‘the center’. And there is a grid of other points drawn on the floor. Periodically you get instructions to move. ‘4 to the east’, ‘2 to the north’ and sometimes something like ‘increase your distance to the center 4 times’
You find that these instructions always lead you to another grid point. If you need to increase the distance to the center you just find your location relative to it e.g. 2 dots west and 4 dots North, and repeat these steps the number of times that is specified.
This plane of grids together with those instructions forms an algebraic ring.
In contrast, if you also allowed instructions like ‘half the distance to the center’. The grid of points would not be enough for you to move around. Standing one dot away from the center, you couldn’t half the distance without leaving the grid. Instead you find that now you can basically move anywhere on the plane. You also cannot leave the plane. There’s no instruction telling you to go up. The set of instructions is again complete in that sense. The plane with those expanded instructions now represents an algebraic field.
Rings are usually assumed to have a multiplicative identity element, but sometimes this is not part of the definition of a ring. (Sometimes rings without identity elements are called “rngs.”) An example is the set of even integers (whose identity element *would* be 1, but 1 is not in the set). Or the set of sequences converging to 0, with pointwise addition and multiplication. The identity would be (1,1,1,…), but that’s not in the set. Fields always have a multiplicative identity.
Rings are not necessarily commutative, so it is not necessarily true that xy = yx for all x, y in the ring. For example, the set of n by n matrices (with elementwise addition and normal matrix multiplication) is not commutative. Fields are always commutative.
Rings may contain zero divisors, nonzero elements x so that xy = 0 for some y. Fields can’t contain zero divisors.
Rings can have proper ideals, fields can’t. That means rings can admit quotients, but fields can’t. Fields are much more rigid.
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