Given two finite dimensional algebra $A$ and $B$ such that $A$ is Gorenstein and $B$ is not. Can the trivial extension algebras of $A$ and $B$ be isomorphic? See http://www.sciencedirect.com/science/article/pii/0022404984900586 1.3. for the definition. Gorenstein means here that the injective dimension of the regular module is finite as a left and as a right module.