I'm having trouble understanding the following definition from my textbook:
If $M$ is an $A$-module and $\phi: M \longrightarrow M$ an $A$-linear endomorphism of $M$, I write $A'[\phi] \subset$ End $M$ for the subring generated by $A$ and the action of $\phi$.
What do the elements of $A'[\phi]$ look like? Can anyone illustrate with an example? And is there a name for this subring? The way I understand it, $A'[\phi]$ consists of all compositions of $\phi$ with elements of $A$ (if we view multiplication by an element $a \in A$ as a map), is that correct?
In general, if $B$ is a ring, $A$ is a subring of $Z(B)$ (center) and $\phi \in B$, then $A[\phi]$ denotes the smallest subring of $B$ containing $A$ and $\phi$. Its elements are polynomial expressions $a_0 + a_1 \phi + \dotsc + a_n \phi^n$ with $n \geq 0$ and $a_i \in A$. Notice that $A[\phi]$ is a quotient of $A[x]$, the polynomial ring in one variable (modulo the ideal of all polynomials which vanish when evaluated at $\phi$). I don't think that there is a special name for this. Borrowing field theory language, we might call it a "simple ring extension of $A$".