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{\Huge 189-235A: Basic Algebra I} \\ \vskip 0.1in
{\Huge Assignment 7} \\ \vskip 0.1in
{\Large Due: Wednesday, November 16 }
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1. Let $F$ be a field, and let $f:F\lra R$ be a homomorphism satisfying
$f(0)\ne f(1)$.
Show that $f$ is necessarily injective, and that the image of $f$ is isomorphic
to $F$.
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2. Let $R$ be the ring $\Z_p[x]$ of polynomials with coefficients in the
finite field $\Z_p$, and let
$f:R \lra S$ be a surjective homomorphism from $R$ to a ring $S$.
Show that $S$ is either isomorphic to $R$,
or is a finite ring.
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Let $R$ be a commutative ring and let $I$ be an ideal of $R$.
Prove or disprove the following statements.
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3. If $R$ is an integral domain, then so is $R/I$.
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4. If every ideal in $R$ is principal, the same is true of $R/I$.
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5. If every ideal in $R/I$ is principal, the same is true of $R$.
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6. Let $R=\Z[x]$, and let $I$ be the ideal $(p,x^2+1)$ generated by the integer
prime $p$ and the polynomial $x^2+1$. Show that $R/I$ is
isomorphic to $\Z/5\Z\times\Z/5\Z$ is $p=5$, and is isomorphic to a field
with $49$ elements if $p=7$.
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7. Let $F$ be a field and let $R=F[[x]]$ denote the ring of formal power series
with coefficients in $F$, i.e., the set of expressions of the form
$$ \sum_{n=0}^\infty a_n x^n,\quad a_n\in F,$$
where the addition and multiplication are performed by
formally expanding out the sums and products
(without worrying about issues of convergence, which
don't make sense in an arbitrary field $F$ anyways!)
Let $I=(x)$ be the ideal generated by the power series $x$. Show that
$R/I$ is isomorphic to $F$. Show that any element of $R$ which does not
belong to $I$ is invertible. Conclude that any non-trivial ideal of $R$ is
contained in $I$.
(A ring with this property is called a {\em local ring}, a terminology
arising from the prototypical example $F[[x]]$, because power series can be
thought of as ``functions defined only around the value $x=0$".)
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8. Using the isomorphism theorem, identify the quotient ring $R/I$ with a more
familiar ring that you already know.
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8a. $R=$ the ring of continuous real-values functions on $\R$;
$I=$ the ideal of functions $f\in R$ satisfying
$f(1)=f(2)=0$.
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8b. $R=\Z[x]$, $I= n \Z[x]$ for some integer $n\in\Z$.
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8c. $R=\R[x]$, $I=(x^2+1)\R[x]$.
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8d. $R=\Z[x]$, $I=(2x-1)\Z[x]$.
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