Math 360, Fall 2016, Assignment 14

I must study politics and war that my sons may have liberty to study mathematics and philosophy.

- John Adams, letter to Abigail Adams, May 12, 1780

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 a ring element).
4. Commutative ring.
5. Unital ring.
6. Unity element (of a unital ring).
7. Unit (in a unital ring).
8. Group of units (of a unital ring; a.k.a. multiplicative group of a unital ring).
9. Left zero-divisor.
10. Right zero-divisor.
11. Integral domain.
12. Field.
13. Trivial ring (a.k.a. zero ring).

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

1. Theorem concerning multiplication by zero.
2. Laws of sign (in rings).
3. Theorem characterizing when $1=0$.

Solve the following problems:

1. Section 18, problems 1, 5, 7, 8, 9, 10, 11, 12, 13, 15, and 20.
2. Section 19, problems 1 and 2 (be careful on 1; since $\mathbb{Z}_{12}$ has zero-divisors, the only way to be sure you've found all solutions is to try every ring element).
3. Give examples of each of the following:

(a) A field.

(b) An integral domain that is not a field.

(c) A commutative, unital ring that is not an integral domain.

(d) A commutative ring that is not unital.

(e) A unital ring that is not commutative.

(f) A ring that is not unital or commutative.

4. (Optional) Show that in $M_n(\mathbb{R})$, every left zero-divisor is also a right zero-divisor, and vice-versa. (Hint: this requires you to remember linear algebra very well, which is why it is optional. First try to decide which matrices are left zero-divisors; your answer should involve kernels. Then try to decide which matrices are right zero-divisors; the answer involves images. Then use the rank-nullity theorem.)

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