An algebra A is left weakly Gorenstein if any semi-Gorenstein-projective left A-modules is Gorenstein-projective. The weakly Gorensteinness of two kinds of algebras are answered. Using the method of the monomorphism category, it is proved that the tensor algebra A B with gl. dim B< is left weakly Gorenstein if and only if so is A. For a class of Morita algebras Λ=pmatrixsmallmatrix A and then it is proved that Λ is left weakly Gorenstein if and only if so are A and B. As an application, the upper triangular matrix algebra Tₙ (A) is left weakly Gorenstein if and only if so is A.
Gao et al. (Thu,) studied this question.