Homework 8

Due: Nov 10th, 2021

Instructions

Please find and use the LaTeX template for homework.

Use LaTeX to create a PDF. Upload your PDF to Gradescope. If you don’t have LaTeX on your computer, you can use Overleaf. Don’t submit the LaTeX source, just the PDF. Email your instructor (that’s me) if you have questions or need help.

Please include your name and the homework number within the document. Some additional formatting instructions are in the syllabus. To summarize:

  • Use a new page (\newpage) for each problem.
  • State which question you are answering and the actual question. Then, start your answer in a new paragraph.
  • Use environments such as proof and theorem (via \begin{proof}...\end{proof}).
  • Use 12pt option \documentclass[12pt]{amsart} and linespacing \linespread{2.4}.

You are encouraged to work together on the homework!

Using a computer for computations

On this homework you are allowed to use a computer to do some of the long computations with Gröbner bases. Take a look at basic directions to help you get started.

Problems to turn in on Gradescope

  1. Exercise 2.8.1.

  2. Exercise 2.8.5.

  3. Exercise 3.1.3.

  4. Try to prove all of the following. Turn in the best three (3) that you are able to prove.

    Suppose \(f : R \to S\) is a homomorphism.

    1. \(f\) is injective (one-to-one) if and only if the kernel of \(f\) is zero, \(\ker f = \{0\}\).

    2. For elements \(r_1,r_2 \in R\), \(f(r_1) = f(r_2)\) if and only if \(r_1 - r_2 \in \ker f\). (In other words, \(r_1 \equiv r_2 \pmod{\ker f}\).)

    3. If \(I \subseteq S\) is any ideal, then \(f^{-1}(I)\) is an ideal. (\(f^{-1}(I)\) is defined to be \(f^{-1}(I) = \{ r \in R \mid f(r) \in I \}\).)

    4. If \(I \subseteq S\) is any ideal, then \(\ker f \subseteq f^{-1}(I)\).

    5. If \(I_1, I_2 \subseteq S\) are any ideals and \(I_1 \subseteq I_2\), then \(f^{-1}(I_1) \subseteq f^{-1}(I_2)\). (We say that \(f^{-1}\) is “inclusion preserving”.)

  5. Try to prove all of the following. Turn in the best three (3) that you are able to prove.

    1. Let \(R\) be a ring (commutative, with \(1\)). Let \(I \subseteq R\) be an ideal. Then \(I = R\) if and only if \(1 \in I.\)

    2. Suppose \(I \subseteq k[x_1,\dotsc,x_n]\) is an ideal. If \(1 \in I\) then \(V(I)\) is empty.

    The next statements are about radical ideals. Let \(R\) be a ring (commutative, with \(1\)). Definition. An ideal \(I \subseteq R\) is called radical if the following condition holds: If \(a \in R\) and there is an \(n \geq 1\) such that \(a^n \in I\), then \(a \in I\).

    1. In \(\mathbb{Z}\) show that the ideal generated by \(6\) is radical and the ideal generated by \(12\) is not radical.

      (Hint: If \(a \in \mathbb{Z}\) and there is an \(n \geq 1\) such that \(a^n\) is divisible by \(6\), then \(a\) is divisible by \(6\).)

    2. Prove that for any variety \(V \subseteq k^n\), the ideal \(I(V)\) is radical.

    3. Prove that the intersection of radical ideals is radical. (We have already shown that the intersection of ideals is an ideal. You don’t need to show again that the intersection is an ideal, just that it’s radical.)

Problems to complete on WebWork

No WebWork problems this week.

Additional reading, required

You don’t need to turn in anything for these tasks, but you do need to do them.

  1. Continue reading Chapter 3 of the textbook.

Additional reading, optional