F-algebra

Section F-algebra

We begin by defining an \(F\)-algebra, where \(F\) is a field.

Definition 19.

Let \(F\) be a field. An \(F\)-algebra or an algebra over a field \(F\) is an associative ring \(A\) with unity together with a ring homomorphism \(\varphi\colon F\to Z(A)\text{.}\) The ring \(A\) is also given an \(F\)-vector space structure by defining the scalar multiplication \(\alpha\cdot a\mapsto \varphi(\alpha)a\text{.}\)

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Observation 20.

Suppose that \(A\) is an \(F\)-algebra. Then, \(A\) is both a ring and an \(F\)-vector space. Furthermore, the ring and vector space structures are compatible, i.e., for \(\alpha\in F\) and \(a,b\in A\) we have the following.

\begin{align*} \alpha\cdot(ab) \amp =\varphi(\alpha)(ab)\\ \amp =(\varphi(\alpha)a)b\\ \amp =(\alpha\cdot a)b \end{align*}

Similarly, we have the following by using the fact that \(\varphi(\alpha)\in Z(A)\text{.}\)

\begin{align*} \alpha\cdot(ab) \amp =\varphi(\alpha)(ab)\\ \amp =(\varphi(\alpha)a)b\\ \amp =(a\varphi(\alpha))b\\ \amp =a(\varphi(\alpha)b)\\ \amp =a(\alpha\cdot b) \end{align*}

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Therefore,

\begin{equation} \alpha\cdot(ab)=(\alpha\cdot a)b=a(\alpha\cdot b)\tag{1} \end{equation}

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Observation 21.

Note that for \(\alpha\in F\) and \(a\in A\) we have \((\alpha\cdot 1_A)a=\alpha\cdot a\text{.}\) Also, \(a(\alpha\cdot 1_A)=\alpha\cdot a\text{.}\) Hence, \((\alpha\cdot 1_A)a=a(\alpha\cdot 1_A)\text{.}\)

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Example 22.

Let \(F\) be a field. Then, the polynomial ring \(F[x]\) is an \(F\)-algebra.
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The ring of \(n\times n\) matrices with entries in \(F\text{,}\) denoted by \(M_n(F)\text{,}\) is an \(F\)-algebra.
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If \(E/F\) is a field extension, then \(E\) is an \(F\)-algebra.
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Definition 23. Algebra homomorphism.

Suppose that \(F\) is a field and that \(A,B\) are \(F\)-algebras. An \(F\)-algebra homomorphism from \(A\) to \(B\) is a ring homomorphism as well as an \(F\)-linear map.

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Example 24. Evaluation as an algebra homomorphism.

Let \(E/F\) be a field extension, and let \(\alpha\in E\text{.}\) The evaluation map \(\ev_\alpha\colon F[x]\to E\) given by \(p(x)\mapsto p(\alpha)\) is an \(F\)-algebra homomorphism.

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