Math 361, Spring 2019, Assignment 1

Read:

1. Section 18.

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

1. Ring.
2. Zero element (of a ring).
3. Opposite (of an element of a ring).
4. Unital ring.
5. Unity element (of a unital ring).
6. Unit (in a unital ring).
7. Division ring.
8. Field.
9. Zero ring.
10. Homomorphism (of rings).
11. Monomorphism.
12. Epimorphism.
13. Isomorphism.
14. Unital homomorphism.
15. Subring.
16. Unital subring.
17. Subfield.
18. Direct product (of two rings).

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

1. Theorem concerning multiplication by zero.
2. Laws of sign.
3. Theorem characterizing the zero ring (in terms of a relation between the zero element and the unity).

Solve the following problems:

1. Section 18, problems 3, 5, 7, 9, 11, 12, 15, 20, 21, 23, 31, and 32.

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