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

  1. \(x+y,-x \in I\)

  2. \(x \cdot y \in I\)

The first condition is equivalent to requiring

\[ x-y \in I \]

We write \(I \triangleleft R\).

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 \(\phi : R \rightarrow R/I\) that takes \(x\) to \(I+x\) is a surjective ring homomorphism that is called the natural map. We have \(ker \phi = I\), 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 \(f:R\rightarrow S\) is a ring homomorphism with kernel \(I\) and image \(C\) then \(R/I \cong C\).

Proposition: Let \(I\triangleleft R\) and \(\phi:R\rightarrow R/I\) be the natural map.

Then the ideals \({\mathcal { I}}\) of \(R/I\) have the form \(\mathcal{I} = J/I = \{I +j | j\in J\}\) for some \(I\subset J\triangleleft R\).

Example: \(\mathbb{Z}/9\mathbb{Z} \cong \mathbb{Z}_9\) has ideals \(\mathbb{Z}/9\mathbb{Z}, 3\mathbb{Z}/9\mathbb{Z}, 9\mathbb{Z}/9\mathbb{Z}\) that correspond under \(\phi^{-1}\) to \(\mathbb{Z} \supset 3\mathbb{Z} \supset 9\mathbb{Z}\), which are all the ideals of \(\mathbb{Z}\) containing \(9\mathbb{Z}\).

Ben Lynn 💡