Zariski topology on Affine spaces, Name of Functor

Question

I've been studying the Zariski topology in my free time.

So I found this functor between Polynomial Algebras and Affine Spaces.

First, we have this $T$ such that for any affine space $(F^n,\mathcal{Z})$ we have: $$T(F^n)=F[x_1,\cdots,x_n]$$ And for any morphism $\phi:F^n\to F^m$ (A function of polynomial coordinates) we have a $F-$algebra homomorphism $T(\phi):T(F^m)\to T(F^n)$ defined by $$T(\phi)(f)=f\circ \phi$$

I know this can be generalized defining morphisms of algebraic sets, not only affine spaces.

So... how is this functor called? I've seen that when we want to build a Zariski topology on the family of all the prime ideals of some commutative rings, we arrive to a similar functor called $\text{Spec}$. In this case, is it called $\text{Spec}$ too? Or does it receive another name?

Answer 1

The possible names are Spm or mSpec or similar, that is the maximal spectrum of a commutative ring with unit; if you don't working in the scheme setting, otherwise $T$ is Spec!