Math 360, Fall 2015, Assignment 13

"Reeling and Writhing, of course, to begin with," the Mock Turtle replied; "And then the different branches of Arithmetic - Ambition, Distraction, Uglification, and Derision."

- Lewis Carroll, Alice's Adventures in Wonderland

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

1. Ring.
2. Commutative ring.
3. Unital ring.
4. Unity (of a unital ring).
5. Integer multiple (of a ring element).
6. Characteristic (of a ring).
7. Zero ring.
8. Left zero-divisor.
9. Integral domain.
10. Unit (of a unital ring).
11. Division ring.
12. Field.

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

1. Theorem concerning multiplication by zero.
2. Theorem characterizing the unital rings in which $1=0$.
3. Laws of sign.
4. Theorem characterizing rings in which $0$ is a unit.

Solve the following problems:

1. Section 18, problems 1, 5, 7, 9, 11, 12, 15, 17, and 19.

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