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Let algebras always be finite dimensional (and connected). In https://arxiv.org/pdf/0809.4897v3.pdf , the algebras with global dimension equal to the dominant dimension equal to one are classified as those algebras Morita equivalent to algebras of upper triangular matrices over divisions rings. Generalising this, one gets to the following problem: What are the algebras with Gorenstein dimension equal to the dominant dimension equal to one? (The Gorenstein dimension equals the global dimension in case the global dimension is finite, so this question can be seen as more general) Alternatively, those are the algebras A where there exists a short exact sequence of the form: $0 \rightarrow A \rightarrow I_0 \rightarrow I_1 \rightarrow 0$, where $I_0$ is projective-injective and $I_1$ is injective but not projective.

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