# 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.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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Then rv, urv [not member of] [K.sub.0], a prime ideal of S, for r [member of] S.
An integral domain is a principal ideal domain (PID) if every prime ideal is principal.
An ideal A is said to be a prime ideal if z * z' [member of] A implies z [member of] A or z [member of] A for all z, z [member of] [Q.sub.t].
Rumynin has defined the restricted universal enveloping algebra of a restricted Lie-Rinehart algebra L in the obvious way, and proved the corresponding Poincare-Birkhoff-Witt theorem in the case that L is projective: In a localization at a prime ideal, the restricted universal enveloping algebra is a free module with a PBW basis truncated at p-th powers.
I is a real ideal of L if I [not equal to] L.A real ideal I of L is a prime ideal if
Thus every prime ideal is semiprime, and PI(L) [subset or equal to] Rd(L).
Clearly every prime ideal is weakly prime and {0} is always weakly prime ideal of N.
(1) X = [Spec.sup.g] (G(R)) = {p< G(R); graded prime ideal may
Separation: Every prime ideal [Mathematical Expression Omitted] generates a prime ideal of analytic functions [Mathematical Expression Omitted].
A neutrosophic soft ideal P over (R, E) is said to be a neutrosophic soft prime ideal if (i) P is not constant neutrosophic soft ideal, (ii) for any two neutrosophic soft ideals M, N over (R, E), MoN [subset or equal to] P [??] either M [subset or equal to] P or N [subset or equal to] P.
The height of a prime ideal p, denoted by height p, is defined by the supremum of integers t such that there exists a chain of prime ideals [mathematical expression not reproducible].
Let [parallel] x [[parallel].sub.[upsilon]] := [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] if [upsilon] is the prime defined by a prime ideal p of [O.sub.k] and [[upsilon].sub.p] is the corresponding valuation.

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