What is Algebra? — Mathematics

Definition 1.33 University Mathematics — Year 2 · Chapter 1 — Sets and Structures

A $K$ -algebra is a $K$ -vector space $A$ with a ring structure whose multiplication is $K$ -bilinear. Examples: $K[X]$ , $\mathcal{M}_n(K)$ , $\mathcal{L}(E)$ , function spaces $\mathcal{F}(X, K)$ , $\C$ as an $\R$ -algebra. Morphisms of algebras are linear ring morphisms; the evaluation $P \mapsto P(u)$ from $K[X]$ to $\mathcal{L}(E)$ (or $\mathcal{M}_n(K)$ ) is the central example, driving Chapter 3.

Examples

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 $A^2 = 0$ ,

P(A) = P(0)\,I + P'(0)\,A = \begin{pmatrix} P(0) & P'(0)\\ 0 & P(0)\end{pmatrix},

(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.

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