Math 361, Spring 2022, Assignment 5

Read:

1. Section 18.
2. Section 19.

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

1. The initial morphism from $\mathbb{Z}$ to any unital ring $R$.
2. $\mathrm{char}(R)$ (the characteristic of a unital ring $R$).
3. The prime subring of a unital ring $R$.
4. Zero-divisor (in a commutative ring $R$).
5. Integral domain.
6. Field.

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

1. Theorem relating the prime subring to the characteristic (i.e. "The prime subring of a unital ring $R$ is an isomorphic copy of...")
2. Formula for $\mathrm{char}(\mathbb{Z}_a\times\mathbb{Z}_b)$.
3. Chinese Remainder Theorem.
4. Theorem concerning the characteristic of an integral domain.

Solve the following problems:

1. Section 18, problems 15, 17, 18, and 40.
2. Section 19, problems 1, 2, 5, 7, 9, and 11.
3. (The Freshman's Dream) Suppose that $R$ is a commutative, unital ring of characteristic two, and choose any $a,b\in R$. Prove that $(a+b)^2=a^2+b^2$. (Please do not reveal this theorem to actual freshmen, who must work in rings of characteristic zero and who already have enough trouble squaring binomials correctly.)
4. (The Freshman's Dream in general) Generalize the above exercise as follows: let $R$ be a commutative, unital ring of prime characteristic $p$, and let $a,b\in R$ be arbitrary. Prove that $(a+b)^p=a^p+b^p$. (Hint: use the binomial theorem, which is valid in any commutative ring.)
5. Give an example to show that the Freshman's Dream does not hold in composite characteristic.
6. Suppose that $R$ is a commutative, unital ring, and that $a\in R$ is a unit. Show that $a$ is not a zero-divisor. (Hint: suppose to the contrary that there exists $b\neq0$ with $ab=0$. What happens if you multiply this equation by $a^{-1}$?)
7. Prove that every field is an integral domain.
8. Generalize the above result by showing that any unital subring of a field is an integral domain. (Hint: Suppose that $F$ is a field and $R$ is a unital subring of $F$. If $R$ had zero-divisors, then they would also be zero-divisors in $F$.)
9. Suppose that $D$ is an integral domain. Show that $\mathrm{char}(D)$ is either zero or a prime. (Hint: suppose to the contrary that $\mathrm{char}(D)$ is composite, say $\mathrm{char}(D)=nm$ for $n,m>1$ and let $\iota$ be the initial morphism. What is $\iota(n)\cdot\iota(m)$?)

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