920 Extra Problems Ring Theory

Ring Theory

Basic Structure

  • Show that if an ideal I ⊴ 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.
problem (Units or Zero Divisors):

Every a\in R for a finite ring is either a unit or a zero divisor.

solution:
\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.
problem (Maximal implies prime):

Maximal \implies prime, but generally not the converse.

solution:

  • 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.
problem (Jacobson radical is bigger than the nilradical):

The nilradical is contained in the Jacobson radical, i.e. \begin{align*} {\sqrt{0_{R}} } \subseteq J(R) .\end{align*}

solution:

Maximal \implies prime, and so if x is in every prime ideal, it is necessarily in every maximal ideal as well.

problem (Mod by nilradical to kill nilpotents):

R/ {\sqrt{0_{R}} } has no nonzero nilpotent elements.

solution:

    

      

\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*}

    

  
problem (Nilradical is intersection of primes):
    

      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*}

    

  
solution:

    
    
      
        
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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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