Ring Theory
Basic Structure
 Show that if an ideal \(I{~\trianglelefteq~}R\) contains a unit then \(I = R\).
 Show that \(R^{\times}\) need not be closed under addition.
Ideals
 \(\star\) Show that if \(x\) is not a unit, then \(x\) is contained in some maximal ideal.
Every \(a\in R\) for a finite ring is either a unit or a zero divisor.
\hfill
 Let \(a\in R\) and define \(\phi(x) = ax\).
 If \(\phi\) is injective, then it is surjective, so \(1 = ax\) for some \(x \implies x^{1}= a\).
 Otherwise, \(ax_1 = ax_2\) with \(x_1 \neq x_2 \implies a(x_1  x_2) = 0\) and \(x_1  x_2 \neq 0\)
 So \(a\) is a zero divisor.
Maximal \(\implies\) prime, but generally not the converse.

Suppose \({\mathfrak{m}}\) is maximal, \(ab\in {\mathfrak{m}}\), and \(b\not\in {\mathfrak{m}}\).

Then there is a containment of ideals \({\mathfrak{m}}\subsetneq {\mathfrak{m}}+ (b) \implies {\mathfrak{m}}+ (b) = R\).

So \begin{align*} 1 = m + rb \implies a = am + r(ab) ,\end{align*} but \(am\in {\mathfrak{m}}\) and \(ab\in {\mathfrak{m}}\implies a\in {\mathfrak{m}}\).
Counterexample: \((0) \in {\mathbf{Z}}\) is prime since \({\mathbf{Z}}\) is a domain, but not maximal since it is properly contained in any other ideal.
 Show that every proper ideal is contained in a maximal ideal
 Show that if \(x\in R\) a PID, then \(x\) is irreducible \(\iff \left\langle{x}\right\rangle{~\trianglelefteq~}R\) is maximal.
 Show that intersections, products, and sums of ideals are ideals.
 Show that the union of two ideals need not be an ideal.
 Show that every ring has a proper maximal ideal.
 Show that \(I{~\trianglelefteq~}R\) is maximal iff \(R/I\) is a field.
 Show that \(I {~\trianglelefteq~}R\) is prime iff \(R/I\) is an integral domain.
 Show that \(\cup_{{\mathfrak{m}}\in {\operatorname{maxSpec}}(R)} = R\setminus R^{\times}\).
 Show that \({\operatorname{maxSpec}}(R) \subsetneq \operatorname{Spec}(R)\) but the containment is strict.
 Show that every prime ideal is radical.
 Show that the nilradical is given by \({\sqrt{0_{R}} } = \sqrt{(}0)\).
 Show that \(\text{rad}(IJ) = \text{rad}(I) \cap\text{rad}(J)\)
 Show that if \(\operatorname{Spec}(R) \subseteq {\operatorname{maxSpec}}(R)\) then \(R\) is a UFD.
 Show that if \(R\) is Noetherian then every ideal is finitely generated.
Characterizing Certain Ideals
 Show that for an ideal \(I{~\trianglelefteq~}R\), its radical is the intersection of all prime ideals containing \(I\).
 Show that \(\sqrt{I}\) is the intersection of all prime ideals containing \(I\).
The nilradical is contained in the Jacobson radical, i.e. \begin{align*} {\sqrt{0_{R}} } \subseteq J(R) .\end{align*}
Maximal \(\implies\) prime, and so if \(x\) is in every prime ideal, it is necessarily in every maximal ideal as well.
\(R/ {\sqrt{0_{R}} }\) has no nonzero nilpotent elements.
\begin{align*} a + {\sqrt{0_{R}} } \text{ nilpotent } &\implies (a+ {\sqrt{0_{R}} })^n \coloneqq a^n + {\sqrt{0_{R}} }= {\sqrt{0_{R}} } \\ &\implies a^n \in {\sqrt{0_{R}} } \\ &\implies \exists \ell \text{ such that } (a^n)^\ell = 0 \\ &\implies a\in {\sqrt{0_{R}} } . \end{align*}
The nilradical is the intersection of all prime ideals, i.e. \begin{align*} {\sqrt{0_{R}} } = \bigcap_{\mathfrak{p} \in \operatorname{Spec}(R)} \mathfrak{p} \end{align*}

\({\sqrt{0_{R}} } \subseteq \cap\mathfrak{p}\):

\(x \in {\sqrt{0_{R}} } \implies x^n = 0 \in \mathfrak p \implies x\in \mathfrak{p} \text{ or } x^{n1}\in\mathfrak p\).

\(R\setminus{\sqrt{0_{R}} } \subseteq \cup_{{\mathfrak{p}}} (R\setminus\mathfrak{p})\):

Define \(S = \left\{{I{~\trianglelefteq~}R {~\mathrel{\Big\vert}~}a^n\not\in I \text{ for any } n}\right\}\).

Then apply Zorn’s lemma to get a maximal ideal \({\mathfrak{m}}\), and maximal \(\implies\) prime.
Misc
 Show that localizing a ring at a prime ideal produces a local ring.
 Show that \(R\) is a local ring iff for every \(x\in R\), either \(x\) or \(1x\) is a unit.
 Show that if \(R\) is a local ring then \(R\setminus R^{\times}\) is a proper ideal that is contained in the Jacobson radical \(J(R)\).
 Show that if \(R\neq 0\) is a ring in which every nonunit is nilpotent then \(R\) is local.
 Show that every prime ideal is primary.
 Show that every prime ideal is irreducible.