Exercises 3.4 Exercises
2.
Give an example of a ring homomorphism of commutative rings such that the inverse image of a maximal ideal need not be a maximal ideal.
Hint.
Consider the inclusion ring homomorphism \(\Z\to\Q\text{.}\)
3.
Show that in a finite commutative ring every prime ideal is a maximal ideal.
Hint.
Let \(A\) be a finite commutative ring, and let \(\mfp\) be a nonzero prime ideal in \(A\text{.}\) Thus \(A/\mfp\) is a finite integral domain, therefore \(A/\mfp\) is a field.
4.
Suppose that \(A\) is a commutative ring in which \(a^n=a\) for every \(a\in A\) and some \(n\in\N\text{.}\) Show that every prime ideal in \(A\) is maximal.
Hint.
Let \(\mfp\) be a prime ideal in \(A\text{.}\) Let \(0\neq\overline{a}\in A/\mfp\text{.}\) We have \(\overline{a}^{n}=\overline{a}\in A/\mfp\text{,}\) i.e., \(\overline{a}\left(\overline{a}^{n-1}-1\right)=0\in A/\mfp\text{.}\) Since \(\mfp\) is prime, \(A/\mfp\) is an integral domain. Thus either \(\overline{a}=0\in A/\mfp\) or \(\overline{a}^{n-1}=1\in A/\mfp\text{.}\)
5.
Let \(A\) be a commutative ring, and let \(\mfa\) be an ideal in \(A\text{.}\) Define
\begin{equation*}
{\rm rad}(\mfa)=\{a\in A:a^n\in\mfa\quad{\text{for some natural number}\;n}\;\text{that may depend on}\;a\}\text{.}
\end{equation*}
Show that \({\rm rad}(\mfa)\) is an ideal in \(A\text{.}\) Further show that if \(\mfp\) is a prime ideal in \(A\) then \({\rm rad}(\mfp)=\mfp\text{.}\)6.
Let \(C[0,1]\) be the ring of all continuous real-valued functions on \([0,1]\text{.}\) Show that the maximal ideals of \(C[0,1]\) are of the following form
\begin{equation*}
\mfm_a=\left\{f\in C[0,1]:f(a)=0\right\}\quad\text{for some}\;a\in [0,1].
\end{equation*}
Hint.
Let \(\mfm\) be a maximal ideal in \(C[0,1]\text{.}\) Assume that \(\mfm\neq\mfm_a\) for any \(a\in [0,1]\text{,}\) i.e., given \(a\in [0,1]\) there exists \(f_a\in\mfm\subset C[0,1]\) such that \(f_a(a)\neq 0\text{.}\) Since \(f_a\) is continuous function there exists an open interval \(U_a\subset [0,1]\) containing \(a\) such that \(f_a(u)\neq 0\) for any \(u\in U_a\text{.}\) In this way we get an open covering of a compact space \([0,1]\text{.}\) Thus there are \(a_1,a_2,\ldots,a_n\in [0,1]\) such that \([0,1]=\bigcup_{i=1}^nU_{a_i}\text{.}\) Since \(\mfm\) is an ideal, the function \(\sum_{i=1}^nf_{a_i}^2\in\mfm\) and this function does not vanish on \([0,1]\text{.}\) Thus \(\sum_{i=1}^nf_{a_i}^2(x)\gt 0\) for any \(x\in [0,1]\text{.}\) Hence \(\tfrac{1}{\sum_{i=1}^nf_{a_i}^2}\in C[0,1]\text{.}\)
7.
Let \(C[0,1]\) be the ring of all continuous real-valued functions on \([0,1]\text{.}\) Let \(a,b\in [0,1]\) be distinct real numbers. Show that
\begin{equation*}
I=\{f\in C[0,1]:f(a)=f(b)=0\}
\end{equation*}
is an ideal of \(C[0,1]\) which is not a prime ideal.
Hint.
Consider functions \(f(x)=x-a\) and \(g(x)=x-b\text{.}\)
Exercise Group.
Let \(A\) be a commutative ring, and let \(I\) be an ideal of \(A\text{.}\) In each of the following cases decide whether or not \(I\) is a prime ideal of \(A\) or not.
