The definition of minimal faithful module in QF-3 algebra Let $\mathcal{U}$ be a k-algebra, over a field k. $\mathcal{U}$ is said to be a QF-3 algebra if it has a unique minimal faithful representation.
The definition of minimal faithful representation is following:
A faithful representation $\mathcal{B}$ of an algebra $\mathcal{U}$ is said to be a minimal faithful representation if deletion of any direct constituent of  $\mathcal{B}$ leaves a nonfaithful representation, that is, if the corresponding space $V$ is the direct sum of
$V_1$ and $V_2$ with $V_2 \not =0$ then $\mathcal{B_1}$ is not faithful.
But in my mind, a minimal faithful module is a faithful module that doesn't have proper submodule faithful. While the first definition looks like a minimal faithful module is a faithful module that doesn't have proper direct summand faithful.
So who can tell me where is wrong? Thank you.
 A: In Anderson and Fuller's chapter on QF-3 rings (page 340 of Rings and categories of modules), you can find this:

Definition: A faithful left (or right) $R$ module $U$ is said to be a  minimal faithful module in case it is isomorphic to a direct summand of every faithful left (respectively right) module. This rather unusual usage of the adjective "minimal" has become accepted in this particular context.

The same definition is given in Tachikawa's Quasi-Frobenius Rings and Generalizations: QF-3 and QF-1 Rings on page 40.
The same definition appears in Lifting Modules: Supplements and Projectivity in Module Theory page 344.
The same definition again here (The citation is a little harder to write out.) With the note that the terminology is "a bit inappropriate."
This is the definition I have seen conventionally. It is not clear at first glance that it is equivalent to the one you gave. If they are equivalent, it must almost certainly be explained somewhere in the literature.
As for your 'question' in the form of "But in my mind, a minimal faithful module is...", the answer is that there is no reason terms are defined as you want them to be. Someone, somewhere used this name and it stuck, and obviously other mathematicians notice the same dissonance that it causes. Your version is also natural, but perhaps not useful in the same context.
