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