# 920 Extra Problems Ring Theory

## 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*}

• $${\sqrt{0_{R}} } \subseteq \cap\mathfrak{p}$$:

• $$x \in {\sqrt{0_{R}} } \implies x^n = 0 \in \mathfrak p \implies x\in \mathfrak{p} \text{ or } x^{n-1}\in\mathfrak p$$.

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

• 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 $$1-x$$ 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 non-unit is nilpotent then $$R$$ is local.
• Show that every prime ideal is primary.
• Show that every prime ideal is irreducible.