prime ideal

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prime ideal

[′prīm ī′dēl]
(mathematics)
A principal ideal of a ring given by a single element that has properties analogous to those of the prime numbers.
References in periodicals archive ?
For x [member of] N, < x > denote the ideal of N generated by x, and P (N) denotes the intersection of all prime ideals of N.
Clearly every prime ideal is weakly prime and {0} is always weakly prime ideal of N.
Here {0, c} is a weakly prime ideal, but not a prime, since [{0, 2}.
Let N be a near-ring and P a weakly prime ideal of N.
Thus AB = 0 and hence P is weakly prime ideal of N.
A well known result that, if M is a non-void m-system of N and I is an ideal of N with I [intersection] M = [empty set], then there exist a prime ideal P [not equal to] N containing I with P [intersection] M = [empty set].
Chapters consider local and reduced rings, and commutative rings in general, as well as the classification of minimal ring extensions, linear systems theory over commutative rings, and the history and summary of asymptotic stability of associated or attached prime ideals.
Since T is weakly Prime, we On Prime, Weakly Prime Ideals in Semigroups 3 have AB [subset or equal to] [AB) [subset or equal to] T or BA [subset or equal to] [BA) [subset or equal to] T, A [subset or equal to] T or B [subset or equal to] T
In commutative semigroup the prime and weakly Prime ideals coincide.
1]), by taking inverse images of graded prime ideals, see [2] and [15].
Biswas [4] studied fuzzy ideals, fuzzy prime ideals of semirings and they defined fuzzy k-ideal and fuzzy prime k-ideals of semirings and characterized fuzzy prime k-ideals of semirings of non-negative integers and determined all its prime k-ideals.