Math 361, Spring 2017, Assignment 6

Carefully define the following terms, then give one example and one non-example of each:

1. Princpial ideal (generated by an element $a$ in a commutative ring $R$).
2. Principal ideal domain (a.k.a. PID).
3. Standard representative (of a coset in the quotient ring $F[x]/\langle m\rangle$ where $F$ is a field and $m$ is a polynomial with coefficients in $F$).
4. Prime ideal.
5. Maximal ideal.

Carefully state the following theorems (you do not need to prove them):

1. Fundamental theorem of ring homomorphisms.
2. Classification of ideals in $F[x]$ ("Every ideal of $F[x]$ is...").
3. Theorem relating prime ideals to integral domains.
4. Theorem relating maximal ideals to fields.

Solve the following problems:

1. Section 27, problems 1, 2, 3, 10, 11, 15, 16, and 17.

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Questions

I having trouble with Section 27, problem 16). Not quite sure where to start. The question is find a prime ideal of $\mathbb{Z} \times \mathbb{Z}$ that is not maximal. I was under the assumption that every prime ideal was also maximal. In problem 1 of Section 27 say find all the prime ideals and all maximal ideals of $\mathbb{Z}_6$ so...the work is... $ 1 = \mathbb{Z_6}$

$(2) = \{0,2,4\}$

$(3) =\{ 0,3\}$

$(6)= \{0\}$

$(2) = \{0,2,4\}$ Are both prime so they are both prime and maximal. $(3) =\{ 0,3\}$

and $(6)= \{0\}$ is neither prime nor maximal.