Homework Assignment 2

Problems to try

Please try to solve these problems, but you do not turn them in. You can also use these results in future problems. I’m happy to talk about these problems (or any others) if you have questions.

• Section 7.4: 7, 8, 10, 14(a,b,c), 16, 26, 37
• Section 7.5: 4
• Section 7.6: 1, 5(a)

1. Additional Problem. Suppose $R$ is a commutative ring with $1\neq 0$ ring. Let $a,b\in R$. Prove that $(a) \subseteq (b)$ if and only if $b$ divides $a$. (See the footnote¹ for the definition of divides.)

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Problems to submit

These are the problems to turn in for a grade. Please write in complete sentences, use correct punctuation, and organize your work. Once your finish these problems, please follow the directions in the Canvas assignment to submit them. If you have any questions (about the math or writing or submission process or anything), please let me know!

1. Section 7.4: 15

   Note that $\mathbb{F}_2$ denotes the field with only two elements, i.e. $\mathbb{F}_2 = \mathbb{Z}/2\mathbb{Z}$, so in this field $1+1 = 0$. When doing this problem, you should find Problem 14(a,b) from Section 7.4 very helpful.

2. Section 7.4: 25

   Possible approach: for $P$ a prime ideal, work in $\overline{R} = R/P$ (which you know is an integral domain), and try to show $\overline{R}$ is a field.

3. Additional Problem. Consider the ideal $I = (x^4-1)$ in the ring $\mathbb{R}[x]$. Using that $\mathbb{R}[x]$ is a PID²,
   1. determine (with proof) all ideals of $\mathbb{R}[x]$ that contain $I$;
   2. draw the lattice of all ideals that contain $I$; and
   3. for each maximal ideal $M$ containing $I$, determine (with proof) which familiar field $\mathbb{R}[x]/M$ is isomorphic to.

   We did a similar problem in class, but please add in all details and fully justify the arguments. Since every ideal of $\mathbb{R}[x]$ is principal, the additional problem to try and Problem 8 from Section 7.4 should be helpful. To prove the isomorphisms in the final part, consider using the evaluation homomorphism together with the first isomorphism theorem.

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Footnotes