Basic Ring Theory - 7

[Tsakanikas Nickos]

We follow the conventions of the previous posts regarding the ring \( \displaystyle A \).

Let \( M \) be a finitely generated \( A \)-module and let \( \mathfrak{a} \) be an ideal of \( A \) contained in the Jacobson Radical \( \mathfrak{R} \) of \( A \). Show that if \( \mathfrak{a}M = M \), then \( M = 0 \).

Furthermore, if \( N \) is a submodule of \( M \), then show the following implication: \[ \mathfrak{a}M + N = M \implies M = N \]

[Papapetros Vaggelis]

Suppose that the first proposition is true.

Then, for the second one, the module \(\displaystyle{M/N}\) is finitely generated and since

\(\displaystyle{a\,M+N=M}\), we get \(\displaystyle{a\,\left(M/N\right)=M/N}\), so,

\(\displaystyle{M/N=\left\{0\right\}\iff M=N}\).

The first one

Let \(\displaystyle{M=\langle{\left\{x_1,...,x_n\right\}\rangle}}\). Then,

\(\displaystyle{x_i=\sum_{j=1}^{n}a_{i\,j}\,x_{j}\,\,,a_{i\,j}\in a\,,1\leq i\,,j\leq n}\), and

\(\displaystyle{\sum_{j=1}^{n}\left(\delta_{i\,j}\,Id-a_{i\,j}\right)\,x_{j}=0}\).

We conclude that there exists \(\displaystyle{x\in A\cap U(A)}\) such that \(\displaystyle{x\,M=\left\{0\right\}}\).

Now, \(\displaystyle{M=x^{-1}\,x\,M=\left\{0\right\}}\).

Note

\(\displaystyle{\forall\,r\in A, \left(r\in J(A)\implies \exists\,(1-r)^{-1}\right)}\).