Math 361, Spring 2020, Assignment 4

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

1. Characteristic (of a unital ring).

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

1. Theorem concerning the characteristic of an integral domain.
2. Chinese Remainder Theorem.
3. Theorem concerning the units of $\mathbb{Z}_n$.

Solve the following problems:

1. Suppose $\mathrm{char}\ R=2$ and $a,b\in R$. Simplify the expression $(a+b)^2$. (Warning: do not encourage such ideas in freshmen, who must work in characteristic zero.)
2. Section 19, problems 11 and 12.
3. Find all units of $\mathbb{Z}_{12}$.
4. Find all units of $\mathbb{Z}_3\times\mathbb{Z}_4$.
5. How are the answers to the last two problems related? Be as explicit as you can.

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