Weakly Gorensteinness of tensor algebras and Morita algebras

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 & N \\ M & B \\ smallmatrixpmatrix, the (semi-)Gorenstein-projective left $Λ$-modules are computed and described; 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.