# Study on the Development of Neutrosophic Triplet Ring and Neutrosophic Triplet Field

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}= I, in the algebraic structure and then combining ‘‘I’’ with each element of the structure with respect to the corresponding binary operation, denoted *. They call it the neutrosophic indeterminate element, and the generated algebraic structure is then termed as a neutrosophic algebraic structure. Some other neutrosophic algebraic structures can be seen as neutrosophic fields [15], neutrosophic vector spaces [16], neutrosophic groups [17], neutrosophic bigroups [17], neutrosophic N-groups [15], neutrosophic semigroups [12], neutrosophic bisemigroups [12], neutrosophic N-semigroups [12], neutrosophic loops [12], neutrosophic biloops [12], neutrosophic N-loop [12], neutrosophic groupoids [12] and neutrosophic bigroupoids [12] and so on.

## 2. Basic Concepts

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

- 1.
- If $\left(N,\ast \right)$ is well defined, i.e., for any $a,b\in N$, one has $a\ast b\in N$.
- 2.
- If $\left(N,\ast \right)$ is associative, i.e., $(a\ast b)\ast c=a\ast (b\ast c)$ for all $a,b,c\in N$.

**Example**

**1.**

**Example**

**2.**

_{10}

_{10}is not a neutrosophic triplet group, nor even a neutrosophic triplet set.

_{10}= {0, 2, 4, 5, 6, 8} is a commutative neutrosophic triplet group, since the law # is well-defined, commutative, associative, and each element belonging to M has a corresponding neutrosophic triplet.

## 3. Neutrosophic Triplet Rings

**Definition**

**5.**

- 1.
- $(NTR,\ast )$ is a commutative neutrosophic triplet group with respect to $\ast $;
- 2.
- $(NTR,\#)$ is well defined and associateve;
- 3.
- $a\#(b\ast c)=(a\#b)\ast (a\#c)$ and $(b\ast c)\#a=(b\#a)\ast (c\#a)$ for all $a,b,c\in NTR$.

**Notations**

**1.**

**Remark**

**1.**

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

**Notations**

**2.**

**Definition**

**9.**

**Example**

**3.**

* | a | b | c |

a | a | a | a |

b | a | b | a |

c | a | a | c |

- (1)
- If there is at least one “a” among x, y, z, then the result is:x * (y * z) = a and (x * y) * z = a, since “a” transforms everything into “a” according to the above table,i.e., a * a = a * b = b * a = a * c = c * a = a.
- (2)
- If there are only b’s, then b * (b * b) = b and (b * b) * b = b.
- (3)
- If there are only c’s, then c * (c * c) = c and (c * c) * c = c.
- (4)
- If there are two b’s and one c, or two c’s and one b, then x * (y * z) = a and (x * y) * z = a,

_{*}, (a, a, b)

_{*}, (a, a, c)

_{*}, (b, b, b)

_{*}, (c, c, c)

_{*}.

# | a | b | c |

a | a | a | a |

b | a | a | a |

c | a | a | a |

**Theorem**

**1.**

- 1.
- 1.$neut\ast \left(a\right)\ast neut\ast \left(b\right)=neut\ast \left(a\ast b\right)$,
- 2.
- $anti\ast \left(a\right)\ast anti\ast \left(b\right)=anti\ast \left(a\ast b\right)$,
- 3.
- $neut\#\left(a\right)\#neut\#\left(b\right)=neut\#\left(a\#b\right)$; and
- 4.
- $anti\#\left(a\right)\#anti\#\left(b\right)=anti\#\left(a\#b\right)$.

**Proof.**

^{#}(a)#neut

^{#}(b)#b = (a#neut#(a))

^{#}(neut#(b)#b) = a#b,

^{#}(a#b)#b = (a#b)#neut

^{#}(a#b) = a#b

^{#}(a)#anti

^{#}(b).

^{#}(a)#anti

^{#}(b)#b = (a#anti

^{#}(a))#(anti

^{#}(b)#b) = (neut#(a)#neut#(b)) = neut#(a#b),

^{#}(a#b).

**Definition**

**10.**

**Definition**

**11.**

- 1.
- $\left(I,\ast \right)$ is a neutrosophic triplet subgroup of $\left(NTR,\ast \right)$; and
- 2.
- For all $x\in I$ and $r\in NTR$, $x\#r\in I$ and $r\#x\in I$.

**Theorem**

**2.**

**Remark**

**2.**

- 1.
- anti*(a) in general are not unique in NTR.
- 2.
- anti#(a) (if they exist for some element a) in general are not unique in NTR.

**Definition**

**12.**

_{2}= ( {0, 2, 4}, *, #) is a commutative neutrosophic triplet ring, where (2

_{#})

^{2}= 2#2 = 0 with 2 ≠ 0, and (4

_{#})

^{2}= 4#4 = 0 with 4 ≠ 0, therefore the non-zero elements 2 and 4 are neutrosophic nilpotent elements in NTR

_{2}.

_{2}.

**Theorem**

**3.**

^{n}), and anti#(a

^{n}) do exist in NTR for an integer n $\ge $ 1 . Let a, neut#(a) be cancellable in NTR. Then

- 1.
- (neut# (a))
^{n}=neut#(a^{n}), - 2.
- (anti#(a))
^{n}=anti#(a^{n})

**Proof.**

^{2})=neut#(a#a)=neut#(a)#neut#(a)=(neut#(a))

^{2}.

^{n−1})=(neut#(a))

^{n−1}, and we need to prove it for n. By Theorem 1(3)

^{n})=neut#(a#a

^{n−1})=neut#(a)#neut#(a

^{n−1})= neut#(a)# (neut#(a))

^{n−1}=(neut#(a))

^{n}.

**Theorem**

**4.**

^{k}), and anti#(a

^{k}) do exist in NTR for an integer k$\ge $ 1. If a is a neutrosophic nilpotent, that is, a

^{n}= 0 for some integer n > 1, then the following are true.

- 1.
- (neut#(a))
^{n}= neut#(0); and - 2.
- (anti#(a))
^{n}= anti*(0).

**Proof.**

^{n}= 0 for some integer n > 1. Using Theorem 3(1) and (2), we have

^{n}=neut#(a

^{n})=neut#(0) and (anti# (a))

^{n}=anti#(a

^{n})=anti#(0).

## 4. Integral Neutrosophic Triplet Domain and Neutrosophic Triplet Ring Homomorphism

**Definition**

**13.**

**Theorem**

**5.**

- 1.
- If neut#(a) and neut#(b) do exist, then neut
_{#}(a)#neut_{#}(b) = 0 implies neut#(a) = 0 or neut#(b) = 0. - 2.
- If anti#(a) and anti#(b) do exist, then anti#(a)#anti#(b) = 0 implies anti#(a) = 0 or anti#(b) = 0.

**Proof.**

^{#}(a) and neut

^{#}(b) belong to NTR.

**Definition**

**14.**

- 1.
- $f\left(a\ast b\right)=f\left(a\right)\oplus f\left(b\right)$, for all $a,b\in NT{R}_{1}$.
- 2.
- $f\left(a\#b\right)=f\left(a\right)\otimes f\left(b\right)$, for all $a,b\in NT{R}_{1}$.
- 3.
- f(neut*(a)) = neut
^{⊕}(f(a)), for all $a\in NT{R}_{1}$. - 4.
- f(anti*(a)) = anti
^{⊕}(f(a)), for all $a\in NT{R}_{1}$.

## 5. Neutrosophic Triplet Fields

**Definition**

**15.**

- 1.
- $\left(NTR,\ast \right)$ is a commutative neutrosophic triplet group with respect to *.
- 2.
- $\left(NTR,\#\right)$ is a neutrosophic triplet group with respect to $\#$.
- 3.
- $a\#(b\ast c)=(a\#b)\ast (a\#c)$ and $(b\ast c)\#a=(b\#a)\ast (c\#a)$ for all $a,b,c\in NTR$.

**Example**

**4.**

_{6}= {0, 2, 3, 4} $\subset $ ${\mathbb{Z}}_{6}$ is a commutative neutrosophic group under multiplication law # modulo 6 (Example 1). The Cayley Table for Law # is as the following:

# | 0 | 2 | 3 | 4 |

0 | 0 | 0 | 0 | 0 |

2 | 0 | 4 | 0 | 2 |

3 | 0 | 0 | 3 | 0 |

4 | 0 | 2 | 0 | 4 |

_{6}as the following table:

* | 0 | 2 | 3 | 4 |

0 | 0 | 0 | 0 | 0 |

2 | 0 | 2 | 0 | 0 |

3 | 0 | 0 | 3 | 0 |

4 | 0 | 0 | 0 | 4 |

- If a = b = c = 0 then a * (b * c) = 0 = (a * b) * c.
- If a = b = c = 2 then a * (b * c) = 2 = (a * b) * c.
- If a = b = c = 3 then a * (b * c) = 3 = (a * b) * c.
- If a = b = c = 4 then a * (b * c) = 4 = (a * b) * c.

_{6}is a commutative neutrosophic group under the law *.

_{5}is satisfied. (Note that * and # are commutative). So (M

_{5}*,#) is NTF.

**Proposition**

**1.**

**Proof.**

**Theorem**

**6.**

**Theorem**

**7.**

**Theorem**

**8.**

- 1.
- If $S$ is a neutrosophic triplet subring $NT{R}_{1}\left(\ast ,\#\right),$ then $f\left(S\right)$ is a neutrosophic triplet subring of $NT{R}_{2}\left(\oplus ,\otimes \right)$.
- 2.
- If $U$ is a neutrosophic triplet subring of $NT{R}_{2,}$ then ${f}^{-1}\left(U\right)$ is a neutrosophic triplet subring of $NT{R}_{1}$.
- 3.
- If $I$ is a neutrosophic triplet ideal of $NT{R}_{2}$, then ${f}^{-1}\left(I\right)$ is a neutrosophic triplet ideal of $NT{R}_{1}$.
- 4.
- If $f$ is onto, and $J$ is an ideal of $NT{R}_{1}$, then $f\left(j\right)$ is an ideal of $NT{R}_{2}$.

**Proof.**

^{⊕}f(a), and f(anti*(a)) = anti

^{⊕}f(a). Therefore, if $f\left(a\right)\in f\left(S\right)$, then neut

^{⊕}f(a) = f(neut*(a))$\in f\left(S\right)$ and, similarly,

_{2}and f is a neutro-homomorphism. But $neu{t}_{\oplus}(f(a))=f(neu{t}_{\ast}(a))\in U,ant{i}_{\oplus}(f(a))=f(ant{i}_{\ast}(a))\in U$, whence $neu{t}_{\ast}(a)\in {f}^{-1}(U),ant{i}_{\ast}(a)\in {f}^{-1}(U).$

_{1}. Then f(a) ∊ I and f(r) ∊ NTR

_{2}. Because f is a neutro-homomorphism and I an ideal of NTR

_{2}, one has:

_{1}. Then:

## 6. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

- Kleiner, I. From Numbers to Ring: The Early History of Ring Theory. In Elemente der Mathematik; Springer: Berlin/Heidelberg, Germany, 1998; Volume 53, pp. 18–35. [Google Scholar]
- Conferences of the Mathematics and Statistics Department of the Technical University of Catalonia. Emmy Noether Course. Available online: https://upcommons.upc.edu/bitstream/handle/2117/81399/CFME-vol-6.pdf?sequence=1&isAllowed=y (accessed on 21 March 2018).
- Connes, A. Introduction to non-commutative differential geometry. In Lectures Notes in Physics; Springer: Berlin/Heidelberg, Germany, 1984; Volume 1111, pp. 3–16. [Google Scholar]
- Connes, A. Non-commutative differential geometry. In Publications Mathematics; Springer: Berlin/Heidelberg, Germany, 1985; Volume 62, pp. 257–360. [Google Scholar]
- Connes, A. The action functional in non-commutative geometry. In Communications in Mathematical Physics; Springer: Berlin/Heidelberg, Germany, 1988; Volume 11, pp. 673–683. [Google Scholar]
- Kaplansky, I. An Introduction to the Differential Algebra; Hermann: Paris, France, 1957. [Google Scholar]
- Kaplansky, I. Fields and Rings, 2nd ed.; The University of Chicago Press: Chicago, IL, USA, 1972; ISBN 0-226-42450-2. [Google Scholar]
- Herstein, I. Wedderburn’s theorem and a theorem of Jacobson. Am. Math. Mon.
**1961**, 68, 249–251. [Google Scholar] [CrossRef] - Smarandache, F. Neutrosophy: Neutrosophic Probability, Set, and Logic; ProQuest Information & Learning: Ann Arbor, MI, USA, 1998; 105p, Available online: http://fs.gallup.unm.edu/eBook-neutrosophics6.pdf (accessed on 21 March 2018).
- Atanassov, K.T. Intuitionistic fuzzy sets. Fuzzy Sets Syst.
**1986**, 20, 87–96. [Google Scholar] [CrossRef] - Zadeh, L.A. Fuzzy sets. In Fuzzy Sets, Fuzzy Logic and Fuzzy Systems: Selected Papers by Lotfi A. Zadeh; World Scientific: River Edge, NJ, USA, 1996; pp. 394–432. [Google Scholar]
- Kandasamy, W.B.V.; Smarandache, F. Some Neutrosophic Algebraic Structures and Neutrosophic N-Algebraic Structures; Hexis: Frontigan, France, 2006; p. 219. [Google Scholar]
- Kandasamy, W.B.V.; Smarandache, F. N-Algebraic Structures and S-N-Algebraic Structures; Hexis: Phoenix, AZ, USA, 2006; p. 209. [Google Scholar]
- Kandasamy, W.B.V.; Smarandache, F. Basic Neutrosophic Algebraic Structures and Their Applications to Fuzzy and Neutrosophic Models; Hexis: Frontigan, France, 2004; p. 149. [Google Scholar]
- Ali, M.; Smarandache, F.; Shabir, M.; Vladareanu, L. Generalization of Neutrosophic Rings and Neutrosophic Fields. Neutrosophic Sets Syst.
**2014**, 5, 9–14. [Google Scholar] - Agboola, A.; Akinleye, S. Neutrosophic Vector Spaces. Neutrosophic Sets Syst.
**2014**, 4, 9–18. [Google Scholar] - Agboola, A.; Akwu, A.; Oyebo, Y. Neutrosophic groups and subgroups. Int. J. Math. Comb.
**2012**, 3, 1. [Google Scholar] - Smarandache, F.; Ali, M. Neutrosophic triplet group. Neural Comput. Appl.
**2018**, 29, 595–601. [Google Scholar] [CrossRef]

# | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|---|

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

1 | 0 | 3 | 6 | 9 | 2 | 5 | 8 | 1 | 4 | 7 |

2 | 0 | 6 | 2 | 8 | 4 | 0 | 6 | 2 | 8 | 4 |

3 | 0 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 |

4 | 0 | 2 | 4 | 6 | 8 | 0 | 2 | 4 | 6 | 8 |

5 | 0 | 5 | 0 | 5 | 0 | 5 | 0 | 5 | 0 | 5 |

6 | 0 | 8 | 6 | 4 | 2 | 0 | 8 | 6 | 4 | 2 |

7 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

8 | 0 | 4 | 8 | 2 | 6 | 0 | 4 | 8 | 2 | 6 |

9 | 0 | 7 | 4 | 1 | 8 | 5 | 2 | 9 | 6 | 3 |

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**MDPI and ACS Style**

Ali, M.; Smarandache, F.; Khan, M. Study on the Development of Neutrosophic Triplet Ring and Neutrosophic Triplet Field. *Mathematics* **2018**, *6*, 46.
https://doi.org/10.3390/math6040046

**AMA Style**

Ali M, Smarandache F, Khan M. Study on the Development of Neutrosophic Triplet Ring and Neutrosophic Triplet Field. *Mathematics*. 2018; 6(4):46.
https://doi.org/10.3390/math6040046

**Chicago/Turabian Style**

Ali, Mumtaz, Florentin Smarandache, and Mohsin Khan. 2018. "Study on the Development of Neutrosophic Triplet Ring and Neutrosophic Triplet Field" *Mathematics* 6, no. 4: 46.
https://doi.org/10.3390/math6040046