The Auslander-Reiten Translation in Submodule Categories

Let $\mathrm{\Lambda }$ be an artin algebra and $S(\mathrm{\Lambda })$ the category of all embeddings $(A\subseteq B)$ where $B$ is a finitely generated $\mathrm{\Lambda }$-module and $A$ is a submodule of $B$. Then $S(\mathrm{\Lambda })$ is an exact Krull-Schmidt category which has Auslander-Reiten sequences. In this manuscript we show that the Auslander-Reiten translation in $S(\mathrm{\Lambda })$ can be computed within the category of $\mathrm{\Lambda }$-modules by using our construction of minimal monomorphisms. If in addition $\mathrm{\Lambda }$ is uniserial then any nonprojective indecomposable object in $\text{Cal}S(\mathrm{\Lambda })$ is invariant under the sixth power of the Auslander-Reiten translation.