Linear operator algebra

Set

 context $X$…left $\mathcal R$-module
 definiendum $\langle \mathrm{Hom}(X,X),+,\cdot,*,\rangle \in L(X,X)$
 context $\langle \mathrm{Hom}(X,X),+,\cdot\rangle \in \mathcal L(X,X)$ context $*:\mathrm{Hom}(X,X)\times \mathrm{Hom}(X,X)\to \mathrm{Hom}(X,X)$
 $v\in M$ $A,B \in \mathrm{Hom}(X,Y)$
 postulate $(A*B)v = A(B v)$

Discussion

Theorem: A linear operator $A:X\to X$ is bijective if it has an inverse in $L(X,X)$.