When I first talked about rings, I said that a ring is an algebraic
abstraction that, in a very loose way, describes the basic nature of integers. A ring is a full abelian group with respect to addition – because the integers
are an abelian group with respect to addition. Rings add multiplication with an
identity – because integers have multiplication with identity. Ring multiplication doesn’t include an inverse – because there is no multiplicative inverse in
But a ring isn’t just the set of integers with addition and multiplication. It’s an abstraction, and there are lots of thing that fit that abstraction beyond the basic realization of the ring of integers. So what are the elements of those
things? They can be pretty much anything – there are rings of topological spaces,
rings of letters, rings of polynomials. But can we use the abstraction of
the ring to create an abstraction of an object that resembles an integer, rather than an abstraction that resembles the set of all integers?
Obviously, the answer is yes, or I wouldn’t be asking it, right?
The answer is something called an ideal. Ideals capture some of the essence of an integer within the set of integers; there are prime ideals that
capture the essence of prime numbers within a ring.
Suppose we have a ring, (A,+,×). We can define a special subset,
R, such that (R,+) is a subgroup of (A,+), and ∀r∈R, ∀a∈A:
r×a∈R – in other words, R is a subgroup of A, and R is closed
over A with respect to multiplication when a member of R is the right operand. If that is true, then R is a right ideal of A.
We can do the same thing again, only require the subset to be closed with respect to multiplication when any member of A is the left operand; that’s called a left ideal.
An two-sided ideal I is a subset of a ring which is both a left ideal and a
right ideal of the ring. A proper ideal is an ideal that is a proper
subset of its ring. In general, when we just say ideal, we mean “two-sided proper ideal”.
The idea of an ideal of a ring is easiest to grasp by looking at
a couple of examples using the integers. This is a lot easier to grasp given an example. We know that the set Z of all integers is a ring using addition and multiplication. The set of all even integers is an ideal, usually written 2Z. Given any member of
2Z, you can multiply it by any integer, and you’ll get a result that’s a member of 2Z. We can say that 2Z is, in some sense, a representation of the number 2 within the ring of integers – it’s the set of values that can be generated from 2 using multiplication. Similarly, 3Z is the set of all integer multiples of 3, and we can say that it’s a representation of the number 3.
The easiest way to see how an ideas works as a sort of prototypical integer is by looking at prime ideals. An ideal I of a commutative ring R is prime if and only if for every a,b∈R, if a×b∈I, then either a∈I or b∈I. That’s an abstract way of saying something that works out to the definition of prime numbers in integers. A number is prime if and only if every multiple of it must be the product of two numbers, at least one of which is a multiple of the prime. So, for example, 7 is prime: you can’t get a multiple of seven except by multiplying something by seven. But 6 isn’t prime: you can multiply 4 by 9 and get 36 – 36 is a multiple of 6, but neither 4 nor 9 are multiples of 6.
That leads us to an equivalent of prime factors of the integers. We know
that in the integers, every integer can be uniquely defined as the product of a collection of prime numbers. Similarly, if you take a ring, R, and the set of prime ideals of that ring, then every ideal of R can be uniquely defined as a product of prime ideals.