Rings & Ideals

Two operations; ideals absorb products.

The idea

A group has one operation. Ordinary arithmetic has two, and they interact: you can add, you can multiply, and multiplication distributes over addition. A ring is that structure kept in the abstract — an abelian group under $+$, an associative multiplication, and distributivity linking them. The integers, the polynomials $F[x]$, the $n \times n$ matrices over a field and the integers mod $n$ are all rings, so anything proved once for rings is proved for all of them.

The subobject worth studying is not the obvious one. A subring is merely a subset closed under both operations, and it is of limited use, because you cannot build a quotient out of it. Quotients are the goal: to work modulo a subset $I$, declaring $a$ and $a'$ the same whenever their difference lies in $I$, that sameness has to survive multiplication. If $a - a'$ lies in $I$ then $ab - a'b = (a - a')b$ must lie in $I$ too — and $b$ here is any element of the ring, not just one from $I$.

So the useful condition is absorption: an additive subgroup $I$ with $ri \in I$ for every $r$ in the ring and every $i \in I$. Such an $I$ is an ideal, and it is precisely what a quotient ring $R/I$ needs in order to exist. The everyday model is the even integers inside $\mathbb{Z}$: an even number times any integer at all is still even.

Ways to work on it

• Walkthrough. Rings, ideals, units, and the ideals of ℤ.
• Practice. Count the units of ℤ/nℤ.
• Hardest. Sort the elements of ℤ/nℤ into units and zero divisors.

Not sure where to start? Take the ten-question placement test.