Let R = Z, the

ring of integers. Define the fuzzy ideals [[mu].sub.1] and [[mu].sub.2] of Z by

It is also well known that Z[[[zeta].sub.n]] + [[[zeta].sup.-1.sub.n]]] is the ring of integers of the nth real cyclotomic field Q([[zeta].sub.n]] + [[zeta].sup.-1.sub.n]]}.

The ring of integers of Q([xi].sub.n + [[xi].sup.-1.sub.n]).

NQR(Z) the neutrosophic quadruple

ring of integers is a neutrosophic quadruple integral domain.

Let Z be the

ring of integers and (Q, *) be the quasigroup given by the following table:

Ichimura, Note on the

ring of integers of a Kummer extension of prime degree.

Example 2.8: Let Z be the

ring of integers. Then <Z [union] J> is the neutrosophic

ring of integers.

In [Z.sub.30] [equivalent to] {0, 1, 2,..., 29] the

ring of integers modulo 30, 25 is a S-idempotent.

Let <Z [union] I> be a neutrosophic

ring of integers and let (F, A) be a soft set over <Z [union] I>.

One may ask whether this is still true if the

ring of integers in Q([square root of d]) is replaced by the

ring of integers in other number fields and this question is at present far from being solved.

Let (Z12 (I), +, *) be a neutrosophic

ring of integers modulo 12 and let S(I) and T(I) be subsets of [Z.sub.12](7) given by S(I)={0, 6, I, 2I, 3I, ..., 11I, 6 + 1, 6 + 21, 6 + 31, ..., 6 + 11I} and T(I)={0, 21, 41, 61, 101}.

Let K/Q be a real quadratic field with class number one and [O.sub.K] be the

ring of integers of K.Put D be the discriminant of K and [epsilon] > 1 be the fundamental unit of K.