Let $B$ be an artin algebra an suppose there is a faithful projective-injective left $B$-module. Moreover, there is a minimal faithful left ideal $Be$ for some idempotent $e\in B$.
1) What does minimal mean? Minimal with respect to what?
2) Does $Be$ have to be indecomposable?
Thanks in advance.
Apparently, "minimal faithful" is meant to mean "minimal among faithful ideals." Let $R$ be some interesting commutative quasi-Frobenius ring (say, $\Bbb Z/(36)$ ). In such a ring, nontrivial ideals have nontrivial annihilators due to the double-annihilator property. This means that the entire ring is a minimal faithful ideal (actually, the minimum faithful ideal) of the ring.
Moreover, the same ring is a faithful projective-injective module over itself, but it is not an irreducible module.