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\parbox{1.95in}{\bf Math 4010}
\hfill {\Large \bf Homework \#1} \hfill
\parbox{1.95in}{\bf \hfill Due: Fri., Aug. 29${}^\mathbf{th}$, 2025}

\vspace{0.2in}

\begin{description}
\item[\#1 An Intersectional Problem:] Intersections play nicely with algebraic things. Unions do not. 

{\it Recall:}
 
\qquad $\bullet$  $x \in \bigcap\limits_{\alpha \in I} S_\alpha$ $\Longleftrightarrow$ for all $\alpha \in I$ we have $x \in S_\alpha$

\qquad $\bullet$  $x \in \bigcup\limits_{\alpha \in I} S_\alpha$ $\Longleftrightarrow$ there exists some $\alpha \in I$ such that $x \in S_\alpha$
\begin{enumerate}[(a)]
\item Let $R$ be a ring and suppose for some index set $I$, we have $S_\alpha$ is a subring of a ring $R$ for all $\alpha \in I$.\\ \force\hfill Show $\bigcap\limits_{\alpha \in I} S_\alpha$ is a subring of $R$.
\item Let $\mathbb{F}$ be a field and suppose for some index set $I$, we have $\mathbb{E}_\alpha$ is a subfield of $\mathbb{F}$ for all $\alpha \in I$.\\ \force\hfill $\bigcap\limits_{\alpha \in I} \mathbb{E}_\alpha$ is a subfield of $\mathbb{F}$. 
\item {\bf[Grad.]} Let $S$ and $T$ be subrings of some ring $R$. Show that $S \cup T$ is a subring if and only if either $S\subseteq T$ or $T \subseteq S$ (i.e., $S\cup T$ is subring only when it equals either $S$ or $T$).
\end{enumerate}

\item[\#2 Calculus! Well, not really:] Let $R$ be a (commutative) ring (with 1) and let $f(x)=r_0+r_1x+\cdots+r_nx^n \in R[x]$. We can define the formal derivative\footnote{This is a totally formal notion of derivative. There is no concept of ``limit'' in a general ring $R$. I will also note that, for example, $2r_2x$ is not $2$ times $r_2x$ but instead it is the $2^{\mathrm{nd}}$ additive power of $r_2x$. In other words, $2r_2x = r_2x+r_2x$. This may not show up in your proof, but it is something we should be aware of.} of $f(x)$ as follows: $f'(x) = \dfrac{d}{dx}\Big[f(x)\Big] =  r_1+2r_2x+\cdots+nr_nx^{n-1}$. 

Compactly: $\dfrac{d}{dx}\left[ \sum\limits_{i=0}^n r_ix^i \right] = \sum\limits_{i=1}^{n} ir_ix^{i-1}$

\vspace{0.05in}

Prove the derivative is a (linear) derivation on $R[x]$. In other words, show (for all $f(x),g(x) \in R[x]$ and $c \in R$):
\begin{itemize}
\item $[f(x)+g(x)]' = f'(x) + g'(x)$
\item $[cf(x)]' = cf'(x)$
\item $[f(x)g(x)]'=f'(x)g(x)+f(x)g'(x)$
\end{itemize}

\vspace{0.05in}

{\it Hint/Note:} When $f(x)=r_0+r_1x+\cdots+r_nx^n$, it is often helpful to adopt the convention that $r_k=0$ for any $k>n$ and $r_k=0$ for any $k<0$. This sometimes simplifies notation in proofs. Also, proving linearity is quite easy. To prove the product rule, you probably will want to use linearity.

Following my hit/note, you probably won't need this fact but as a reminder:\\ \force\hfill If $f(x) = \sum\limits_{i=0}^n r_ix^i$ and $g(x) = \sum\limits_{j=0}^m s_jx^j$, then $f(x)g(x) = \sum\limits_{\ell=0}^{m+n} \left(\sum\limits_{k=0}^\ell r_ks_{\ell-k}\right)x^\ell$.

\item[\#3 Degrees of Difficulty:] Let $R$ be a commutative ring with 1 and for $0\not=f(x) \in R[x]$ let $\partial(f(x)) = \mathrm{deg}(f(x))=$ the degree of $f(x)$.
   \begin{enumerate}[(a)]
   \item Let $R$ be an integral domain, $f(x),g(x) \in R[x]$, and $f(x),g(x) \not=0$. Briefly explain why the leading coefficient of $f(x)g(x)$ is the product of the leading coefficients of $f(x)$ and $g(x)$. Then justify why $\partial(f(x)g(x)) = \partial(f(x)) + \partial(g(x))$. Use this to prove: If $R$ is an integral domain, then so is $R[x]$.
   \item Consider $R=\mathbb{Z}_4[x]$. Show that $(2x+1)^2=1$. What does this say about the results of part (a)?
   \item {\bf [Grad.]} Show that $x$ can be factored: $x=f(x)g(x)$ in $\mathbb{Z}_4[x]$ in such a way that neither $f(x)$ nor $g(x)$ is constant. Can this happen in $R[x]$ when $R$ is an integral domain?
   \end{enumerate}
   
\item[\#4 Polynomial Units:] Recall, for a ring $R$, we let $R^\times = \{ r \in R \;|\; r^{-1} \text{ exists}\}$ (i.e., the group of units).
   \begin{enumerate}[(a)]
   \item Let $R$ be an integral domain. Show $(R[x])^\times = R^\times$. 
   
   Also, what is $(\mathbb{Z}[x])^\times$? What is $(\mathbb{F}[x])^\times$ when $\mathbb{F}$ is a field?
   \item Give an example of an infinite ring whose only unit is $1$.
   \item Give an example of a non-constant polynomial in $\mathbb{Z}_4[x]$ that is a unit.\footnote{This shows that part (a) can break when our coefficient ring is not a field.}
   \end{enumerate}

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