in which H is a residually finite dimensional Hopf algebra, H[degrees] is its finite dual Hopf algebra, which is dense in H*, and L is a dense

subring of En[d.

Then NQS is called a neutrosophic quadruple

subring of NQR if (NQS, +,.

1) [LAMBDA] a set of indices, A a solid

subring of the ring [K.

Then the ring [mathematical expression not reproducible] is a commutative

subring of [mathematical expression not reproducible].

A

subring I [subset] E is called an algebraic ideal of E if

c] [less than or equal to] End A be the

subring of R generated by the commutator actions:

1) For n = 2k [greater than or equal to] 4 and positive integer i, j satisfied k [greater than or equal to] i [greater than or equal to] 1, k [greater than or equal to] j [greater than or equal to] i and i [not equal to] 1 when j = k, any

subring of [M.

Lemma 3: Let B be a

subring of A with 1 [member of] B such that B satisfies A.

An intuitionistic fuzzy subset A of R is said to be an intuitionistic fuzzy

subring of R (IFSR) if it satisfies the following axioms:

o](M), the ring of continuous real-valued functions on M, [Mathematical Expression Omitted], the

subring of bounded continuous functions, and [C.

We claim that Z is a Nash manifold, and that I is a real ideal, so that, by the real Nullstellensatz, the ring D = A[z]/I is a

subring of the ring N (Z) of Nash functions of Z.

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