The almost Gorenstein property appeared in the work of Barucci and Fr�oberg in the context of 1-dimensional analytical unramified rings. It was extended to 1- dimensional local rings by Goto, Matsuoka and Thi Phuong, and later on to rings of higher dimension by Goto, Takahashi and Taniguchi. Let R be a positively graded Cohen-Macaulay K-algebra with canonical module ωR. We let a = − min{k ∈ Z : (ωR)k 6= 0}, which is also known as the a-invariant of R. R is called (graded) almost Gorenstein if there exists an exact sequence of graded R-modules 0 → R → ωR(−a) → E → 0, where E is an Ulrich module. Let H ⊂ N d be a normal affine semigroup, R = K[H] its semigroup ring over the field K and ωR its canonical module. The Ulrich elements for H are those h in H such that for the multiplication map by x h from R into ωR, the cokernel is an Ulrich module. In this talk, we provide some algebraic criterion for testing the Ulrich property. This is based on a joint work with J�urgen Herzog and Dumitru I. Stamate.
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