Alperin-Glauberman replacement operation from arbitrary submodule to invariant submodule

Definition
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 $$L$$

By idempotent we mean that applying the mapping twice is equivalent to applying it once.

To define this map, we proceed in several steps: 

We are now in a position to define the star operation. For a $$R$$-submodule $$M$$ of $$L$$, first consider the submodule generated by its image under $$\sigma$$, i.e., define:

$$S(M) := \langle \sigma(x) \mid x \in M\rangle$$

Now, we define:

$$M^* := \{ 0 \} \cup \{ C(f) \mid f \in S(M) \}$$

Then, $$M^*$$ is the image of $$M$$ under the Alperin-Glauberman replacement operation.



Related facts

 * Basic proposition on Alperin-Glauberman replacement operation for submodules
 * Basic proposition on Alperin-Glauberman replacement operation for subalgebras