What is a ring in the context of abstract algebra?
In: Mathematics
A ring is a set of things that you can add and multiply together, according to the following laws:
for any `a`, `b`, and `c` in the ring,
* `a+(b+c) = (a+b)+c = a+b+c` (associative addition)
* `a + b = b + a` (commutative addition)
* `a * (b + c) = a * b + a*c` (left-distributivity, right-distributivity is similar)
* `a + 0 = 0 + a = a` (additive identity)
* `a*(b*c) = (a*b)*c = a*b*c` (associative multiplication)
* `a*1 = 1*a = a` (multiplicative identity)
Note that multiplication is not assumed to be commutative. Some authors don’t require that rings contain a multiplicative identity.
Notable examples of rings are Z (the integers) and quotients of Z, namely Z/nZ, where the result of every operation is taken modulo n i.e. take the remainder after dividing by n.
Basically an abelian group with an additional operation. That new operator is distributive over the original operator.
Think addition over integers for the abelian group and add multiplication with association/distribution.
I’m addition to the other answers, which treat your question very rigorously, I think an intuitive answer is appropriate too. Rings are mostly about *not* assuming division will work.
Rings are somewhat of a generalization of polynomials. We can add and subtract polynomials in very nice ways, and we can divide them but there’s no guarantee that the quotient of two polynomials will be another polynomial.
You could also think of rings as a generalization of matrices. Addition and subtraction work really well, but in some ways it doesn’t even make sense to ask about division.
If we want to study structures where we have one operation that we can undo and one operation that we may not necessarily be able to undo, a ring is just that. If the multiplication is reversible, then you have a field. When is multiplication reversible? Textbooks have been written trying to answer this question.
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