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.
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