Let $A$ be a finite dimensional algebra. A NCR of an algebra $A$ is a faithful module $M$ such that $End_A(M)$ has finite global dimension. Call a faithful module PNCR (=pseudo-NCR) in case $End_A(M)$ has Cartan determinant equal to one.
While checking whether $M$ is PNCR takes mostly only seconds with a computer, checking whether $M$ is NCR is usually impossible with a computer as it takes too long.
Call $A$ great in case $M$ is a NCR if and only if $M$ is a PNCR. For example $A=K[x]/(x^n)$ is great.
Question 1: Are there other nontrivial great algebras?
Question 2: Are Brauer tree algebras great? (The Brauer tree algebras with 2 simples and multiplicity one are also great, but higher dimensional cases take too long with the computer aleady).
