Math 361, Spring 2020, Assignment 3

Read:

1. Section 19.

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

1. Zero-divisor.
2. Integral domain.
3. Field.
4. Initial morphism (to a unital ring).
5. Prime subring (of a unital ring).

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

1. Cancellation law (in an integral domain).
2. Theorem relating unital subrings of fields to integral domains.
3. Fundamental Theorem of (Ring) Homomorphisms.

Solve the following problems:

1. Section 19, problems 1, 2, 3, 6, 7, 8, and 9. (Note: the characteristic of a unital ring is the non-negative generator of the kernel of the initial morphism. So for problems 6-9, begin by trying to understand the initial morphism and computing its kernel. We will have much more to say about the characteristic of a ring next week.)
2. Describe the prime subring of $M_2(\mathbb{Z}_3)$. (Hint: first make a table of values for the initial morphism.) Is the prime subring all of $M_2(\mathbb{Z}_3)$?
3. Describe the prime subring of $\mathbb{Z}_3\times\mathbb{Z}_3$. Is it all of $\mathbb{Z}_3\times\mathbb{Z}_3$?
4. Describe the prime subring of $\mathbb{Z}_2\times\mathbb{Z}_3$. Is it all of $\mathbb{Z}_2\times\mathbb{Z}_3$?
5. Prove that $\mathbb{Z}_2\times\mathbb{Z}_3$ is isomorphic to $\mathbb{Z}_6$. (Hint: use the Fundamental Theorem.)
6. Prove that $\mathbb{Z}_3\times\mathbb{Z}_3$ is not isomorphic to $\mathbb{Z}_9$. (Hint: compute the characteristic of each ring.)

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