On Generalizations of Prime Ideals (II) |
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Authors: | M Ebrahimpour |
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Institution: | 1. Department of Mathematics, Faculty of Sciences , Vali-e-Asr University , Rafsanjan , Iran m.ebrahimpour@vru.ac.ir |
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Abstract: | Let R be a commutative ring with identity. We say that a proper ideal P of R is (n ? 1, n)-weakly prime (n ≥ 2) if 0 ≠ a 1…a n ∈ P implies a 1…a i?1 a i+1…a n ∈ P for some i ∈ {1,…, n}, where a 1,…, a n ∈ R. In this article, we study (n ? 1, n)-weakly prime ideals. A number of results concerning (n ? 1, n)-weakly prime ideals and examples of (n ? 1, n)-weakly prime ideals are given. Rings with the property that for a positive integer n such that 2 ≤ n ≤ 5, every proper ideal is (n ? 1, n)-weakly prime are characterized. Moreover, it is shown that in some rings, nonzero (n ? 1, n)-weakly prime ideals and (n ? 1, n)-prime ideals coincide. |
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Keywords: | Quasi-local ring (n ? 1 n)-Weakly prime ideal (n ? 1 n)-Prime ideal |
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