What is Ideal?

Example 1.27 (A polynomial gcd, two ways)

Compute $\gcd(X^3 - 1,\ X^2 - 1)$ in $\Q[X]$. By Euclid:

so the gcd is $X - 1$, and back-substitution gives the Bézout relation

By ideals: the ideal $(X^3 - 1)\Q[X] + (X^2 - 1)\Q[X]$ is principal (Theorem 1.26); it contains $X - 1$ (the display) and is contained in $(X - 1)\Q[X]$ (both generators vanish at $1$, hence are multiples of $X - 1$): the monic generator is $X - 1$. The closing insight: the ideal viewpoint identifies the gcd without dividing — common roots locate the ideal, and Euclid merely certifies it.

Example 1.34 (An evaluation morphism and its kernel)

Take $A = \begin{pmatrix}0 & 1\\ 0 & 0\end{pmatrix}$ and the evaluation $\varepsilon_A \colon \R[X] \to \mathcal{M}_2(\R)$, $P \mapsto P(A)$. Since $A^2 = 0$,

(only the constant and linear terms of $P$ survive). Hence $\ker\varepsilon_A = \{P : P(0) = P'(0) = 0\} = X^2\,\R[X]$: a principal ideal, exactly as Theorem 1.26 predicts, generated by the monic $X^2$ of least degree in the kernel — the minimal polynomial of $A$, star of Chapter 3. The image is the two-dimensional commutative algebra $\{aI + bA\}$: evaluation morphisms shrink the infinite-dimensional $\R[X]$ onto small, computable algebras.

Example 2.20 (A ring without unique factorization)

None of the implications Euclidean $\Rightarrow$ PID $\Rightarrow$ UFD is an equivalence, and the failure of the last is worth seeing once in complete detail. In

the norm is multiplicative and $N(z) = 1$ iff $z \in A^\times = \{\pm1\}$. Consider

All four factors are irreducible: their norms are $4, 9, 6, 6$, and a proper factorization $z = z_1z_2$ would force $N(z_1) \in \{2, 3\}$ — but $a^2 + 5b^2$ never equals $2$ or $3$ ($b = 0$ leaves the non-squares $2, 3$; $\abs b \geq 1$ gives $\geq 5$). Yet $2$ is associate to neither $1 \pm \iu\sqrt5$ (norms $4 \neq 6$): two genuinely different factorizations of $6$ into irreducibles. Equivalently, irreducible $\neq$ prime here: $2$ divides the product $(1 + \iu\sqrt5)(1 - \iu\sqrt5) = 6$ but neither factor (norms again). The ideal-theoretic repair of this failure — factorizing ideals rather than elements — is the birth of algebraic number theory; at our level, the example calibrates how special the Euclidean rings $\Z$, $K[X]$, $\Z[\iu]$ of this chapter really are.