Commutative Algebra

Ideals

An ideal of a ring $R$ is a nonempty subset $I \subset R$ such that for all $x, y \in I$

$x + y, - x \in I$
$x \cdot y \in I$

The first condition is equivalent to requiring

x - y \in I

We write $I ◃ R$ .

Since $I$ is an additive subgroup of $R$ , we can form the quotient group

R / I = {I + a ∣ a \in R}

which is the group of cosets of $I$ with addition: for $a, b \in R$ we have

(I + a) + (I + b) = I + (a + b)

It is easily verified that $R / I$ is in fact a ring by defining multiplication as follows:

(I + a) \cdot (I + b) = I + (a \cdot b)

(It needs to be checked that multiplication is well-defined: if $I + a = I + a', I + b = b'$ , then it can be seen that $a b - a' b' = a (b - b') + (a - a') b' \in I$ .)

We call $R / I$ a quotient ring.

The mapping $ϕ : R \to R / I$ that takes $x$ to $I + x$ is a surjective ring homomorphism that is called the natural map. We have $\ker ϕ = I$ , so every ideal is the kernel of some ring homomorphism. The converse is easily verified, that is, the kernels of ring homomorphisms with domain $R$ are precisely the ideals of $R$ .

The following are easy to verify:

Fundamental Homomorphism Theorem: If $f : R \to S$ is a ring homomorphism with kernel $I$ and image $C$ then $R / I ≅ C$ .

Proposition: Let $I ◃ R$ and $ϕ : R \to R / I$ be the natural map.

Then the ideals $ℐ$ of $R / I$ have the form $ℐ = J / I = {I + j ∣ j \in J}$ for some $I \subset J ◃ R$ .

Example: $ℤ / 9 ℤ ≅ ℤ_{9}$ has ideals $ℤ / 9 ℤ, 3 ℤ / 9 ℤ, 9 ℤ / 9 ℤ$ that correspond under $ϕ^{- 1}$ to $ℤ \supset 3 ℤ \supset 9 ℤ$ , which are all the ideals of $ℤ$ containing $9 ℤ$ .