Math 361, Spring 2019, Assignment 2

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. Image (of a ring homomorphism).
4. Kernel (of a ring homomorphism).
5. Initial morphism (from the integers to a given unital ring).
6. Prime subring (of a unital ring).
7. Characteristic (of a unital ring).

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

1. Zero-product property (of integral domains).
2. Cancellation law (in integral domains).
3. Theorem relating fields to integral domains.
4. Theorem concerning when $\mathbb{Z}_n$ is an integral domain, and when it is a field.
5. Theorem concerning images and preimages of subrings.
6. Theorem relating images to epimorphisms.
7. Theorem relating kernels to monomorphisms.
8. Theorem concerning the characteristic of an integral domain.
9. Chinese Remainder Theorem.

Solve the following problems:

1. Section 19, problems 1, 2, 3, 5, 6, 7, 8, 9, and 11.

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