the difference between a ring and a group, and examples of both (other than numbers)
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Rings *are* groups but they have additional structure.
The standard example is the set ℤ of integers with addition and multiplication. (Yeah, that’s numbers.)
So taking any abelian (i. e. commutative) group with an operation noted “+” you can make it a ring if you can define a second operation, noted multiplicatively with “.”, which:
* is associative, i. e. **a.(b.c) = (a.b).c**, to avoid awkwardness in computing products;
* plays well with the already existing addition by being distributive: **a.(b + c) = a.b + a.c** (and (b + c).a = b.a + c.a if your multiplication isn’t commutative).
That’s all there is to it.
Now for examples “other than numbers”:
* the set P(A) of all subsets of a set A with “addition” being the symmetric difference and “multiplication” just the usual intersection;
* the ring of polynomials ℝ[x] over the reals ℝ in one indeterminate x, with the usual high-school algebra rules. You can even replace ℝ with any ring;
* square matrices of fixed size (say nxn) with their coefficients taken from ℝ, ℂ or any ring, addition and multiplication being the usual matrix operations;
* given any abelian group G, you can define the ring of endomorphisms of G (functions f: G —> G compatible with the group structure). Addition is pointwise, (f + g)(x) = f(x) + g(x) and multiplication is function composition, (g . f) (x) = g ( f (x) ).
Note that a ring with a unit 1 (1 . a = a . 1 = a) is called unital and if every nonzero element also has an inverse (a . a^-1 = a^-1 . a = 1) then it is called a field.
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