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In this paper, we concern the Principal Ideal Theorem (PIT) for w-Noetherian rings. Let R be a w-Noetherian ring and a be a nonzero nonunit element of R. If p is a prime ideal of R minimal over (a), then ht p ≦ 1.

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In this paper, we concern the w-analogue of Matijevic’s result. We show that if R is a w-Noetherian ring and T a w-overring of R such that TR wg. Then T has ACC on regular w-ideals.

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