The F-Signature and Strong F-Regularity

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This is a revised version, incorporating referee's comments, of an article published in Mathematical Research Letters, 10 (2003), no. 1, 51-56.

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Observations on the F-signature of local rings of characteristic p (2006, methodology): “Remark 5.8. In [ AL ], I. Aberbach and G. Leuschke dene the s-dimension of (R;m), denoted by sdim(R), to be the largest integeri such that lim supe!1 #(”
Quantifying singularities with differential operators (2018, background): “ for I= m, J= n. For the second statement, the ﬁrst part of the previous proposition applies. Since direct summands of regular rings are strongly F-regular [HH89], they also have positive F-signature [AL03]. We show a similar result for diﬀerential signature. Theorem 6.15. Let (R,m,K) be an algebra ”
The structure of F-pure rings (2003, background): “This result was first established in [2] and our remark is along the lines of the argument given there.”
F-signature and Hilbert–Kunz multiplicity : a combined approach and comparison (2016, unknown): “Lastly, note that Theorem 6.3 can also be applied outside the context of descending chains of irreducible ideals, and gives yet another perspective on the original proof of the existence of the F -signature.”