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\parbox{1.95in}{\bf Math 4720/5210}
\hfill {\Large \bf Homework \#2} \hfill
\parbox{1.95in}{\bf \hfill Due: Mon., Feb. 5${}^\mathbf{th}$, 2024}

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\item[\#1 An Ideal Question:] Let $R$ be a commutative ring with ideals $I$ and $J$. Define $IJ = \{ \sum_{i=1}^n a_ib_i \;|\; n \geq 0 \text{ with } a_1,\dots,a_n \in I \text{ and } b_1,\dots,b_n \in J\}$ where we understand that the empty sum (i.e., $n=0$) is $0$. Show $IJ$ is an ideal.

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{}{\bf [Grad.]} Give an example of a ring $R$ and ideals $I$ and $J$ such that $S=\{ ab \;|\; a \in I \text{ and } b \in J\}$ (just products not sums of products) is {\it not} an ideal. 
{\it Note:} You will need to use non-principal ideals to pull this off.

\item[\#2 Unprincipaled:] The end of our factorization handout claims $I = (x,2) = \{ f(x)x+g(x)2 \;|\; f(x),g(x) \in \mathbb{Z}[x] \}$ (this is the set of all integral polynomials with even constant term) is {\it not} a principal ideal of $\mathbb{Z}[x]$. 

Show this directly.  Is this a prime ideal? Is this a maximal ideal? Explain your answers.

\item[\#3 Prime and Maximal:] As a quick reminder, in $\mathbb{Z}$ and in $\mathbb{Z}_n$, we know that subgroup = normal subgroup = cyclic subgroup = subring = ideal = principal ideal.
\begin{enumerate}[(a)]
\item Find all the ideals of $\mathbb{Z}_{24}$ and draw the corresponding lattice. Which ideals are prime? Which are maximal?
\item Draw the lattice for $\mathbb{Z}_{24}/(6)$. Which are prime? Which are maximal?
\end{enumerate}

\item[\#4 Quotient Issues:] Let $\varphi:R \to S$ be a ring homomorphism and $I \triangleleft R$. We know $J=\varphi(I) = \{ \varphi(x) \;|\; x \in I \} \triangleleft S$. 

Prove $\overline{\varphi}:\diagquotient{R}{I} \to \diagquotient{S}{J}$ ``defined by'' $\overline{\varphi}(r+I) = \varphi(r)+J$ is a {\it well-defined} homomorphism.

In the case, $\varphi$ is an isomorphism, show that $\overline{\varphi}$ is too.
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