Tight Closure

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Throughout this paper all rings are commutative, with identity, and Noetherian, unless otherwise specified. We will summarize many of the results in [H-H] concerning the theory of tight closure and prove several basic theorems using this theory in characteristic p, including the theorem of Briançon-Skoda that the integral closure of the nth power of an n-generator ideal of a regular ring is contained in the ideal, the monomial conjecture, the syzygy theorem, and that summands of regular rings are Cohen-Macaulay (C-M).