5 problems

Takahashi's conjecture. Then $R$ is Cohen–Macaulay.

Let $R$ be a non-Gorenstein local ring, and suppose that there is a nonfree totally reflexive $R$-module. The Takahashi conjecture. There are infinitely many isomorphism classes of…

Let $R$ be a local ring, and let $M$ be a nonzero cyclic $R$-module of finite G-injective dimension. Cyclic module conjecture. If such an $M$ exists, then $R$ is Gorenstein. This c…

Let $R$ be a commutative Noetherian local ring and let $M$ be a finitely generated $R$-module. Assume that $M$ has finite length and that $\operatorname{Gdim}(M)$, the Gorenstein d…

Let $R$ be a commutative Noetherian local ring and let $M$ be a finitely generated $R$-module. Write $\operatorname{Gdim}(M)$ for the Gorenstein dimension,…