# Wikipedia's definition of finitely generated algebra.

This is Wikipedia's definition of a finitely generated algebra:

A finitely generated algebra (also called an algebra of finite type) is an associative algebra $$A$$ over a field $$K$$ where there exists a finite set of elements $$a_1,…,a_n$$ of $$A$$ such that every element of $$A$$ can be expressed as a polynomial in $$a_1,…,a_n$$, with coefficients in $$K$$.

My question is, shouldn't the last part "with coefficients in $$K$$" read "with coefficients in $$\alpha(K)$$" where $$\alpha : K \rightarrow A$$ is the homomorphism that makes $$A$$ a $$K$$-algebra?

• It is customary to assume the product is directly by the scalars of $\;K\;$ , just as with linear spaces (and an algebra over $\;K\;$ is a linear space over $\;K\;$) . If you want to be very formal and the definition mentions that homomorphism, then I guess you could do what you mention at the end....But observe that many define "algebra over a field" as a linear space over that field that is also a ring. This level of formality is usually more than enough. – DonAntonio Jun 23 at 11:54

## 1 Answer

It depends on what is meant. Both are actually valid, in the following sense : if you're looking completely internally to $$A$$, then it makes more sense to say $$\alpha (K)$$.

However, if you have a more global view on all $$K$$-algebras, then you see that for any $$K$$-algebra $$A$$ and any polynomial $$P\in K[X_1,...,X_n]$$, there is an associated map $$A^n\to A$$, and you can say that $$(a_1,...,a_n)$$ generate $$A$$ if for any $$a\in A$$ there exists $$P\in K[X_1,...,X_n]$$ (so a polynomial with coefficients in $$K$$) such that the image of $$(a_1,...,a_n)$$ under the associated map is $$a$$.

The thing is that this associated map is in some sense "independent" of the algebra $$A$$, and it completely determines $$P$$ if you allow yourself to look at all $$K$$-algebras, so it makes sense to say that it is a polynomial with coefficients in $$K$$.

(PS : I'll add my personal opinion here, but I think that's not the best definition of finitely generated algebra; to me the subalgebra of $$A$$ generated by a subset $$S$$ should be the intersection of all subalgebras of $$A$$ containing $$S$$, so literally "the smallest subalgebra containing $$S$$", and then the description with polynomials should be a theorem/characterization, not the definition)