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*}

    

  

    
    
      
        
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      \({\sqrt{0_{R}} } \subseteq \cap\mathfrak{p}\):
    
    
  
        
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      \(x \in {\sqrt{0_{R}} } \implies x^n = 0 \in \mathfrak p \implies x\in \mathfrak{p} \text{ or } x^{n-1}\in\mathfrak p\).
    
    
  
        
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      \(R\setminus{\sqrt{0_{R}} } \subseteq \cup_{{\mathfrak{p}}} (R\setminus\mathfrak{p})\):
    
    
  
        
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      Define \(S = \left\{{I{~\trianglelefteq~}R {~\mathrel{\Big\vert}~}a^n\not\in I \text{ for any } n}\right\}\).
    
    
  
        
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      Then apply Zorn’s lemma to get a maximal ideal \({\mathfrak{m}}\), and maximal \(\implies\) prime.
    
    
  
        
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Misc

    
    
      
        
  • 
          Show that localizing a ring at a prime ideal produces a local ring.
        
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          Show that \(R\) is a local ring iff for every \(x\in R\), either \(x\) or \(1-x\) is a unit.
        
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          Show that if \(R\) is a local ring then \(R\setminus R^{\times}\) is a proper ideal that is contained in the Jacobson radical \(J(R)\).
        
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          Show that if \(R\neq 0\) is a ring in which every non-unit is nilpotent then \(R\) is local.
        
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          Show that every prime ideal is primary.
        
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          Show that every prime ideal is irreducible.
        
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