Math 360, Fall 2015, 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. Subring.
2. Unital subring.
3. Subfield.
4. (Ring) homomorphism.
5. (Ring) monomorphism.
6. (Ring) epimorphism.
7. (Ring) isomorphism.
8. Left ideal.
9. Right ideal.
10. Ideal.
11. Quotient (of a ring by an ideal).
12. Canonical projection (of a ring onto a quotient ring).

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

1. Theorem concerning images and preimages of subrings under homomorphisms.
2. Theorem relating kernels to ideals.
3. Fundamental theorem on (ring) homomorphisms.
4. Corollary relating images of homomorphisms to quotients by kernels.
5. Corollary concerning epimorphisms.

Solve the following problems:

1. Section 19, problems 1 (be careful; the safest method is to try all possible $x$), 2, 5, 7, and 9.
2. Section 26, problems 3, 4, 11, 12, 13, 14, 15, and 22.

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