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.