Math 360, Fall 2017, 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

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 a ring element).
4. Commutative ring.
5. Unital ring.
6. Unity element (of a unital ring).
7. Unit (in a unital ring).
8. Field.
9. Trivial ring (a.k.a. zero ring).
10. Homomorphism (of rings).
11. Monomorphism.
12. Epimorphism.
13. Isomorphism.
14. Structural property.
15. Image (of a homomorphism).
16. Kernel (of a homomorphism).
17. Subring.
18. Ideal.
19. Quotient (of a ring by an ideal).
20. Canonical projection (from $R$ to $R/I$).

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 $1_R=0_R$.
4. Theorem relating injectivity to kernels.
5. Theorem relating kernels to ideals.
6. Theorem characterizing when coset multiplication is well-defined.

Solve the following problems:

1. Section 18, problems 1, 3, 5, 7, 9, 11, 12, 13, 14, 15, 16, 17, and 19.

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