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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Solutions:

Complete Notes:

https://drive.google.com/file/d/1biJbwmyG2FefR-eiZ0QOFmcEOG8aDLud/view?usp=sharing

Definitions:

1. Zero-divisor: Let $R$ be a ring, and $a \in R$. We say that $a$ is a left zero-divisor (may not be commutative) if 1. $a \neq 0$, and 2. $\exists b \in R,$ with $b\neq 0$ but $ab = 0$.
2. Integral domain: An integral domain is a commutative, unital ring, not the zero ring, which has no zero-divisor.
3. Field:A field is a commutative, unital ring, not the zero ring, in which every non-zero element is a unit.
4. Subring: Suppose $R$ is a ring, and $S \subseteq R$. We say that $S$ is a subring of $R$ if: 1. $0_R \in S$ 2. $a,b \in S \Rightarrow a+b \in S$ 3. $a \in S \Rightarrow -a \in S$ 4. $a. b \in S \Rightarrow ab \in S$
5. Unital Subring: A unital subring of a unital ring $R$ is a subring which contains $1_R$. This is not the same as a subring which happens to be unital ($1_S$ might be different.
6. Zero (a.k.a. trivial) subring: Let $R$ be any ring, $S = \{ 0_R\}$ is a subring. The zero subring or the trivial subring. This is the smallest subring.
7. Improper subring: et $R$ be any ring, $S = R$ is a subring. The improper subring. This is the largest subring.
8. Subring generated by a subset.
9. Prime subring (of a unital ring): Suppose $R$ is any unital ring. The prime subring of $R$ is the subring generated by $1_R$. This is the smallest unital subring.

Theorems:

1. Zero-product property (of integral domains): if $D$ is an \textbf{integral domain}, and $a, b \in D$ with $ab = 0$, then either $a = 0$ or $b = 0$.
2. Cancellation law (in integral domains): Suppose $D$ is a domain, $a \neq 0$, and $ab = ac$. Then $b = c$.
3. Theorem relating fields to integral domains: Every field is an integral domain.
4. Theorem characterizing the units and zero-divisors of $\mathbb{Z}_n$: Suppose $[a] \in \mathbb Z$ and $[a] \neq 0$. Then, 1. If $gcd(a,n) = 1$, then $[a]$ is a unit of $\mathbb Z_n$. 2. If $gcd(a,n) \neq 1$, then $[a]$ is a zero-divisor of $\mathbb Z_n$.
5. Theorem characterizing when $\mathbb{Z}_n$ is a field, and when it is an integral domain: If $n$ is prime, then $\mathbb Z_n$ is a field. If $n$ is composite, then $\mathbb Z_n$ is not even an integral domain.

Problems:

Answer to problems:

https://drive.google.com/file/d/1ieuGx7P7BYDkEANSxwteao4HwZjswYVx/view?usp=sharing

Java Program for the book problems 1, 2, 3, 4:

https://drive.google.com/file/d/1qhd2Kw4f1v164GG3qfh5Q9nVqbGA59xe/view?usp=sharing