Affine Algebraic Sets - D&F Chapter 15, Section 15.1

[Math Amateur]

I am reading Dummit and Foote Ch 15, Commutative Rings and Algebraic Geometry. In Section 15.1 Noetherian Rings and Affine Algebraic Sets, Example 2 on page 660 reads as follows: (see attachment)

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(2) Over any field k, the ideal of functions vanishing at $$(a_1, a_2, ... ... ... a_n) \in \mathbb{A}^n$$ is a maximal ideal since it is the kernel of the surjective ring homomorphism from $$k[x_1, x_2, ... ... x_n]$$ to the field k given by evaluation at $$(a_1, a_2, ... ... ... a_n)$$.

It follows that $$I((a_1, a_2, ... ... ... a_n)) = (x - a_1, x - a_2, ... ... ... , x - a_n)$$

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I can see that $$(x - a_1, x - a_2, ... ... ... , x - a_n)$$ gives zeros for each polynomial in $$k[ \mathbb{A}^n ]$$ - indeed, to take a specific example involving $$\mathbb{R} [x,y]$$ we have for, let us say, a particular polynomial $$g \in \mathbb{R} [x,y]$$ where g is as follows:

$$g(x,y) = 6(x - a_1)^3 + 11(x - a_1)^2(y - a_2) + 12(y - a_2)^2$$

so in this case, clearly $$g(a_1, a_2) = 0$$ ... ... ... and, of course, other polynomials in $$\mathbb{R} [x,y]$$ similarly.

BUT ... ... I cannot understand D&Fs reference to maximal ideals. Why is it necessary to reason about maximal ideals.

Since I am obviously missing something, can someone please help by explaining what is going on in this example.

Would appreciate some help.

Peter

Note - see attachment for definition of I(A)[Note: This has also been posted on MHF]

 

[jvicens]

Dear Peter,

Thank you for your question about Example 2 on page 660 of Dummit and Foote's "Commutative Rings and Algebraic Geometry." I can understand why you may be confused about the reference to maximal ideals in this example. Let me try to explain it to you.

First, let's review the definition of a maximal ideal. An ideal is a subset of a ring that is closed under addition and multiplication by elements of the ring. A maximal ideal is an ideal that is not contained in any other proper ideal (i.e. an ideal that is not a subset of any other ideal, except the whole ring itself). In other words, a maximal ideal is an ideal that cannot be "extended" any further within the ring.

Now, in the context of affine algebraic sets, we can think of the ideal of functions vanishing at a point as a "special" ideal. This ideal is generated by polynomials that have the point as a root. So, in the example given, the ideal I((a_1, a_2, ... , a_n)) is generated by the polynomials (x - a_1, x - a_2, ... , x - a_n). These polynomials have the point (a_1, a_2, ... , a_n) as a root, which means that they vanish at that point.

Now, the reason why this ideal is called a maximal ideal is because it cannot be "extended" any further within the ring of polynomials. In other words, there is no other ideal in the ring that contains this ideal as a proper subset. This can be seen by considering the homomorphism from k[x_1, x_2, ... , x_n] to the field k given by evaluation at (a_1, a_2, ... , a_n). This homomorphism maps every polynomial in the ideal I((a_1, a_2, ... , a_n)) to 0, which means that the ideal is "maximal" in the sense that it cannot be extended any further without losing this property.

I hope this explanation helps to clarify the concept of maximal ideals and their relevance in this example. Let me know if you have any further questions.