Math 361, Spring 2022, Assignment 3

Read:

1. Section 19.

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

1. Zero-divisor.
2. Integral domain.
3. Field.
4. Subring.
5. Zero (a.k.a. trivial) subring.
6. Improper subring.
7. Subring generated by a subset.
8. Prime subring (of a unital ring).

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

1. Zero-product property (of integral domains).
2. Cancellation law (in integral domains).
3. Theorem relating fields to integral domains.
4. Theorem characterizing the units and zero-divisors of $\mathbb{Z}_n$.
5. Theorem characterizing when $\mathbb{Z}_n$ is a field, and when it is an integral domain.

Solve the following problems:

1. Section 19, problems 1, 2, 3, 4, and 14 (hint for 14: to show that this matrix is a left zero-divisor, find another matrix whose image in contained in the kernel of this one; to show that it is a right zero-divisor, find another matrix whose kernel contains the image of this one).
2. Describe the prime subrings of $\mathbb{Q}$, of $\mathbb{R}$, and of $\mathbb{C}$.
3. Describe the prime subring of $\mathbb{Z}$.
4. Describe the prime subring of $\mathbb{Z}_n$.
5. Working in the field $\mathbb{Z}_3$, solve the equation $x^3=x$.
6. Working in the field $\mathbb{Z}_5$, solve the equation $x^5=x$.
7. Working in the field $\mathbb{Z}_7$, solve the equation $x^7=x$.
8. By now you probably have a conjecture about $\mathbb{Z}_{11}$. Do not try to prove this. Instead, prove the conjecture for $\mathbb{Z}_p$ where $p$ is an arbitrary prime. (Hint: the conjecture is obviously true if $x=0$. Otherwise $x$ is an element of the group of units of $\mathbb{Z}_p$ (why?). But as we have seen, Lagrange's Theorem implies that in any group $G$ we have $g^{\left\lvert G\right\rvert}=e$ for every $g\in G$. This gives rise to a certain identity for non-zero elements of $\mathbb{Z}_p$. Multiplying both sides of this identity by $x$ will prove the conjecture.)
9. Show by a simple counterexample (e.g. in $\mathbb{Z}_6$) that the result above is not generally true in $\mathbb{Z}_n$ when $n$ is composite. Exactly which part of your proof above breaks in the composite case?
10. Try to correctly generalize the conjecture to the composite case (i.e. formulate and prove a statement which encompasses the prime case but is also true in the composite case). In doing this you will be following in the footsteps of Leonhard Euler; this result (like many others) is known as Euler's Theorem, and it is in fact the mathematical basis of RSA encryption.

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Definitions:

1. Zero-divisor: Let $R$ be a ring, and $a \in R$. We say that $a$ is a \textbf{left zero-divisor} (may not be commutative) if

\begin{enumerate} \item $a \neq 0$, and \item $\exists b \in R, $ with $b\neq 0$ but $ab = 0$.

1. Integral domain.
2. Field.
3. Subring.
4. Zero (a.k.a. trivial) subring.
5. Improper subring.
6. Subring generated by a subset.
7. Prime subring (of a unital ring).