# Download Neutrosophic Rings by Florentin Smarandache W. B. Vasantha Kandasamy PDF

By Florentin Smarandache W. B. Vasantha Kandasamy

Examine on algebraic constitution of crew earrings is among the best, such a lot sought-after themes in ring conception. the recent type of neutrosophic earrings outlined during this e-book shape a generalization of team earrings and semigroup earrings. The examine of the sessions of neutrosophic staff neutrosophic earrings and S-neutrosophic semigroup neutrosophic jewelry which shape one of those generalization of staff jewelry will throw gentle on staff earrings and semigroup earrings that are crucial substructures of them. A salient function of this workforce is the numerous urged difficulties at the new sessions of neutrosophic jewelry, options of so that it will definitely improve the various nonetheless open difficulties in team jewelry. extra, neutrosophic matrix jewelry locate functions in neutrosophic types like Neutrosophic Cognitive Maps (NCM), Neutrosophic Relational Maps (NRM), Neutrosophic Bidirectional stories (NBM) etc.

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**Additional info for Neutrosophic Rings**

**Example text**

S(x) where both r(x) and s(x) are neutrosophic polynomials; if p(x) = r(x) s(x) but only one of r(x) or s(x) is a neutrosophic polynomial then we say p(x) is only semi neutrosophic reducible. If p(x) = r(x) s(x) where r(x) = I or 1 and s(x) = p(x) or I s(x) = p(x) then we call p(x) a irreducible neutrosophic polynomial. 4: Let [〈R ∪ I〉] [x] be a neutrosophic polynomial ring. Let I be an ideal of [〈R ∪ I〉][x], if I is generated by an irreducible neutrosophic polynomial, then we call I the principal neutrosophic ideal of [〈R ∪ I〉] [x].

R ∪ I 〉 , is a neutrosophic ring then we If the quotient ring J 〈R ∪ I 〉 call it as the pseudo quotient neutrosophic ring. If is J just a ring we call it a false pseudo quotient neutrosophic ring. We just give an example of the same. 25: Let 〈Z6 ∪ I〉 be a neutrosophic ring. Let P = {0, I, 2I, 3I, 4I, 5I} be the pseudo neutrosophic ring. The quotient ring 〈 Z6 ∪ I〉 = {P, 1 + P, 2 + P, 3 + P, 4 + P, 5 + P} P is a false pseudo neutrosophic ring. 39 Now we just define one small notion before we proceed to define some more quotient.

Clearly (2 + 2I) I = 0 is a trivial neutrosophic zero divisor. (2 + 2I)2 = 0 is a non trivial neutrosophic zero divisor of 〈Z4 ∪ I〉. (2 + 3I) (2 + 2I) = 0 is a non trivial neutrosophic divisor of zero; 2I. 2 = 0 is also a non trivial neutrosophic divisor of zero, which we name differently. (2 + 2I) . 2 = 0 is also a non-trial neutrosophic divisor of zero, which will be defined differently. 14: Let 〈R ∪ I〉 be a neutrosophic ring x = a + bI, be a neutrosophic element of R (a ≠ 0 and b ≠ 0). If y ∈ R is such that x y = y x = 0 then y is called the semi neutrosophic divisor of zero.