Math 361, Spring 2020, Assignment 1

Imagination does not breed insanity. Exactly what does breed insanity is reason. Poets do not go mad, but chess-players do. Mathematicians go mad, and cashiers; but creative artists very seldom.... The poet only desires exaltation and expansion, a world to stretch himself in. The poet only asks to get his head into the heavens. It is the logician who seeks to get the heavens into his head. And it is his head that splits.

- G. K. Chesterton, Orthodoxy

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. Zero ring.
8. Direct product (of two rings).
9. $M_n(R)$ (the ring of $n\times n$ matrices with entries in a ring $R$).
10. $\mathrm{PolyFun}(R,R)$ (the ring of polynomial functions from a ring $R$

to itself).

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

1. Fundamental theorem of group homomorphisms.
2. Theorem concerning multiplication by zero.

Solve the following problems:

1. Section 18, problems 3, 5, 7, 9, 11, 12, 14, 15, 16, 17, 19, 20, and 31.

(Hint for 31: try $\mathbb{Z}_n$ for various values of $n$.)

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