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{\bf \Large 250A Homework 9}\\[7mm]
{\small Due Monday December 6}\\[4mm]
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\noindent
{\bf Question 1} (DF 7.1.15) 
Rings $R$ where $a^2=a$ $\forall$ $a\in R$ are called Boolean.
Prove that Boolean rings are abelian and $a+a=0$ $\forall$ $a\in R$.

\vspace{.4cm}\noindent
{\bf Question 2} 
Let $R$ be a commutative ring and call $x\in R$ nilpotent if $x^n=0$ for some
$n\in {\mathbb N}$. Prove that the set of nilpotent elements is
an ideal $I$ (i.e. an additive subgroup closed under left and right
multiplication). Show that the quotient $R/I$ has no non-trivial 
nilpotent elements. Give an example of a ring in which the set of
nilpotent elements is not an ideal.


\vspace{.4cm}\noindent
{\bf Question 3}
Let $R$ be a ring and $f,g:{\mathbb Q}\rightarrow R$
be ring homomorphisms coinciding on the integers, i.e. $f|_{\mathbb
Z}=g|_{\mathbb Z}$. Show that $f=g$.

 
\vspace{.4cm}\noindent
{\bf Question 4} 
Give examples of rings $R$ where (i) $R$ has a left ideal that is
not a right ideal (ii) $R$ has zero divisors but $R/I$ does not
for some ideal $I$.


 

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