Generalizations of Prime Ideals |
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Authors: | D D Anderson Malik Bataineh |
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Institution: | 1. Department of Mathematics , The University of Iowa , Iowa City, Iowa, USA dan-anderson@uiowa.edu;3. Department of Mathematics , The University of Iowa , Iowa City, Iowa, USA |
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Abstract: | Let R be a commutative ring with identity. Various generalizations of prime ideals have been studied. For example, a proper ideal I of R is weakly prime (resp., almost prime) if a, b ∈ R with ab ∈ I ? {0} (resp., ab ∈ I ? I 2) implies a ∈ I or b ∈ I. Let φ:?(R) → ?(R) ∪ {?} be a function where ?(R) is the set of ideals of R. We call a proper ideal I of R a φ-prime ideal if a, b ∈ R with ab ∈ I ? φ(I) implies a ∈ I or b ∈ I. So taking φ?(J) = ? (resp., φ0(J) = 0, φ2(J) = J 2), a φ?-prime ideal (resp., φ0-prime ideal, φ2-prime ideal) is a prime ideal (resp., weakly prime ideal, almost prime ideal). We show that φ-prime ideals enjoy analogs of many of the properties of prime ideals. |
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Keywords: | Almost prime ideal Prime ideal φ-Prime ideal Weakly prime ideal |
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