the difference between a ring and a group, and examples of both (other than numbers)


the difference between a ring and a group, and examples of both (other than numbers)

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Not 100x certain but a ring can describe several groups.

For instance you may have a whole drug dealing ring. But its ultimately just a whole bunch of independent groups working roughly under a banner.

Its not like if theres a pedophilia ring getting busted by the police that its one group.. likely they found one guy and this guy lead to many other groups.. and for the sake of it we call it a ring

Thats why you have things you can call ringleaders ultimately..its just one guy with connection to many other groups… in many cases a ringleader is unaware hes even a ringleader i suppose.

A ring is essentially a commutative group along with another group-like structure that is associative and has an identity element. I’m not really enough of an expert to give you a non-numerical example.

One intuitive way to think about groups is that they are sets of symmetries of things. For example, if I have a square, I can rotate it by 90 degrees and it will look the same as if I hadn’t rotated it at all – so rotation by 90 degrees is an element of the group of symmetries of a square, among other things.

The most important example of a group is the set of permutations of n things- all possible reorderings of {1,2,3,…,n}. It contains n! elements and is called the symmetric group, and it’s a theorem that every group of size n is contained within the symmetric group of size n.

Importantly, groups only have one “operation” (way of combining elements), usually called multiplication (but sometimes addition).

A ring is a group but with some additional structure(or alternately, a field with fewer requirements). Namely, a ring is an abelian group under addition (ie, a+b=b+a for all a, b in the ring). It also has a multiplication which has to be associative and distribute over addition. Unlike a field, the multiplication need not commute, and you don’t require multiplicative inverses. Examples include matrices under matrix addition and matrix multiplication, and polynomials, among many others.

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.