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{\bf \Large 250A Homework 10}\\[7mm]
{\small Ungraded, solutions to appear soon...}\\[4mm]
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\noindent
{\bf Question 1} 
({\it Internal direct sum/product})
Show that if a ring $R=J_1+\cdots+J_n$ and the ideals $J_i\cong R_i$
for rings $R_i$ and $J_i\cap\sum_{j\neq i} J_j=\{0\}$, then
$R\cong\otimes_{i=1}^n R_i$.

\vspace{.4cm}\noindent
{\bf Question 2} 
Look up Zorn's lemma on partially ordered sets ({\it e.g.} Dummit and Foote
Appendix I). Let $R$ be a ring with 1 and at least one proper ideal (an ideal 
other than $R$ or $\{0\}$). Now prove that $R$ has a maximal proper ideal
({\it i.e.} not contained in any other proper ideal)\footnote{Hint: study
``chains'' in $\{J|I\subseteq J\subseteq R\}$ for some proper ideal $I$.}.


\vspace{.4cm}\noindent
{\bf Question 3}
{\it Frobenius homomorphism}
Let $R$ be a commutative ring with identity and prime characteristic $p$.
Show that the map
\be
\varphi:R\longrightarrow R,\, , \quad r\longmapsto r^p\, ,
\ee
is a ring homomorphism.
 
\vspace{.4cm}\noindent
{\bf Question 4} 
Find all ideals in ${\mathbb Z}$.


 

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