Basic proposition on Alperin-Glauberman replacement operation for submodules

Statement
Suppose $$R$$ is an associative unital ring and $$L$$ is a finitely generated right module over $$R$$. Suppose $$\alpha$$ is an automorphism of $$L$$ as a $$R$$-module such that there exists a natural number $$n$$ such that:


 * 1) $$(\alpha - 1)^n(x) = 0$$ for all $$x \in L$$
 * 2) The natural numbers $$1,2,3,\dots,n-1$$ are invertible in $$R$$.

The Alperin-Glauberman star operation is an explicitly defined idempotent mapping $${}^*$$:

All $$R$$-submodules of $$L$$ $$\stackrel{*}{\to}$$ $$\alpha$$-invariant submodules of $$L$$

The basic proposition establishes the above:


 * 1) The image $$M^*$$ of any module $$M$$ is also a module.
 * 2) The image $$M^*$$ of any module $$M$$ is $$\alpha$$-invariant.
 * 3) $$(M^*)^* = M^*$$

Proof of (1)
Note that by definition, $$S(M)$$ is already a right $$R$$-module (since it is defined as the submodule generated by a set). $$M^*$$ is defined as the union of zero and the set of leading coefficients of $$S(M)$$. The proof thus follows from the fact that the set of leading coefficients of a submodule of the polynomial module is itself a module. This can be shown in a manner similar to the Hilbert basis theorem.