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\centerline{\bf HOMEWORK \#4, DUE 10/28}
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\centerline{\bf MATH 504A}
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\noindent 1. Show that any square matrix $M$ over any field is similar to its transpose $M^t$.
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\noindent 2. Show that, if the square matrices $M,N$ over the field $K$ are similar over some larger field $L$ (that is, conjugate via an invertible matrix over $L$), then they are already similar over $K$.
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\noindent 3. Classify the finitely generated projective modules over a PID $R$.
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\noindent 4. Let $M$ be a free module of rank $n$ over a commutative ring $R$.  Show that the $k$th
exterior power $\bigwedge^k M$ is also free over $R$ and determine its rank, by exhibiting an explicit basis for it in terms of a basis $b_1,\ldots,b_n$ of $M$.  Use determinants to show that your basis is linearly independent over $R$.
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\noindent 5. Show that any $\Bbb Z$-module $M$ injects (there is an injective from $M$) into an injective $\Bbb Z$-module $N$.
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