I am reading Dummit and Foote (D&F) Section 15.1 on Affine Algebraic Sets.

On page 662 (see attached) D&F define a morphism or polynomial map of algebraic sets as follows:

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Definition. A map is called a morphism (or polynomial map or regular map) of algebraic sets if

there are polynomials such that

for all

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D&F then go on to define a map between the quotient rings k[W] and k[V] as follows: (see attachment page 662)

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Suppose F is a polynomial in .

Then is a polynomial in

since are polynomials in .

If , then for every

since .

Thus

It follows that induces a well defined map from the quotient ring

to the quotient ring :

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My problem is, how exactly does it follow (and why?) that induces a well defined map from the quotient ring to the quotient ring ?

Can someone (explicitly) show me the logic of this - why exactly does it follow?

Peter