Abstract.
For a reduced F-finite ring R of characteristic p>0 and q=pe one can write 
where M
q
has no free direct summands over R. We investigate the structure of F-finite, F-pure rings R by studying how the numbers a
q
grow with respect to q. This growth is quantified by the splitting dimension and the splitting ratios of R which we study in detail. We also prove the existence of a special prime ideal 
(R) of R, called the splitting prime, that has the property that R/(R) is strongly F-regular. We show that this ideal captures significant information with regard to the F-purity of R.
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References
Aberbach, I.M.: Extension of weakly and strongly F-regular rings by flat maps. J. Algebra 241, 799–807 (2001)
Aberbach, I.M., Enescu, F.: When does the F-signature exist?. Ann. Fac. Sci. Toulouse Math. To appear
Aberbach, I.M., Leuschke, G.: The F-signature and strong F-regularity. Math. Res. Lett. 10(1), 51–56 (2003)
Cowden Vassilev, J.: Test ideals in quotients of F-finite regular local rings. Trans. Am. Math. Soc. 350(10), 4041–4051 (1998)
Enescu, F.: F-injective rings and F-stable primes. Proc. Am. Math. Soc. 131, 3379–3386 (2003)
Fedder, R.: F-purity and rational singularity. Trans. Am. Math. Soc. 278, 461–480 (1983)
Glassbrenner, D.: Strong F-regularity in images of regular rings. Proc. Am. Math. Soc. 124(2), 345–353 (1996)
Hochster, M.: Cyclic purity versus purity in excellent Noetherian rings. Trans. Am. Math. Soc. 231(2), 463–488 (1977)
Hochster, M., Huneke, C.: Tight closure and strong F-regularity. Memoire no. 38, Soc. Math. France, 1989, pp. 119–133
Hochster, M., Huneke, C.: F-regularity, test elements, and smooth base change. Trans. Am. Math. Soc. 346(1), 1–62 (1994)
Hochster, M., Huneke, C.: Applications of the existence of big Cohen-Macaulay algebras. Adv. Math. 113(1), 45–117 (1995)
Hochster, M., Roberts, J.L.: The purity of the Frobenius and local cohomology. Adv. Math. 21(2), 117–172 (1976)
Huneke, C., Leuschke, G.: Two theorems about maximal Cohen-Macaulay modules. Math. Ann. 324(2), 391–404 (2002)
Kunz, E.: On Noetherian rings of characteristic p. Am. J. Math. 98(4), 999–1013 (1976)
Watanabe, K., Yoshida, K.: Minimal relative Hilbert-Kunz multiplicity. Illinois J. Math. 48(1), 273–294 (2004)
Yao, Y.: Observations on the F-signature of local rings of characteristic p>0. Preprint, 2003
Yao, Y.: Modules with finite F-representation type. J. London Math. Soc. To appear
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Dedicated to Professor Melvin Hochster on the occasion of his sixtieth birthday
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Aberbach, I., Enescu, F. The structure of F-pure rings. Math. Z. 250, 791–806 (2005). https://doi.org/10.1007/s00209-005-0776-y
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DOI: https://doi.org/10.1007/s00209-005-0776-y