Recommended Problems

The following problems from Dummit and Foote review the basic ideas on modules and their homomorphisms.

Problem 5.

If $M$ is an $R$ module and $I$ is a left ideal of $R$, define

\[IM = \lbrace \sum_{\mathrm{finite}} a_i m_i : a_i\in I, m_{i}\in M \rbrace\]

Prove that $IM$ is a submodule of $M$.

Problem 8.

Given a ring $R$ and an $R$-module $M$, let

\[\Tor(M)=\lbrace m\in M : \exists r\in R, rm=0, r\not=0 \rbrace\]

Prove that, if $R$ is an integral domain, then $\Tor(M)$ is a submodule of $M$.

Show that if $R$ is not an integral domain then $\Tor(M)$ need not be a submodule of $M$.

Problem 9.

If $N$ is a submodule of an $R$-module $M$, defined the annihilator of $N$ in $R$ to be the set of $r\in R$ such that $rn=0$ for all $n\in N$.

Show that the annihilator of $N$ is a two-sided ideal of $R$.

Problem 19.

Let $V=\R^{2}$ and let $T$ be the linear transformation given by projection onto the $y$-axis. Viewing $V$ as an $\R[x]$ module with $x$ acting as $T$, show that the only proper $\R[x]$ submodules of $V$ are the $x$ and $y$ axes.

Section 10.2

Problem 8

Let $\phi:M\to N$ be a module homomorphism. Prove that $\phi(\Tor(M))\subset \Tor(N)$.

Problem 10

If $R$ is commutative with $1$, prove that $\Hom_{R}(R,R)$ is isomorphic to $R$.

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