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Correction

# Correction: Ali, M., et al. Study on the Development of Neutrosophic Triplet Ring and Neutrosophic Triplet Field. Mathematics 2018, 6, 46

1
Department of Mathematics, Süleyman Demirel University, 32260 Isparta, Turkey
2
Department of Mathematics, University of New Mexico, 705 Gurley Avenue, Gallup, NM 87301, USA
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(6), 565; https://doi.org/10.3390/math7060565
Submission received: 5 May 2019 / Revised: 19 June 2019 / Accepted: 19 June 2019 / Published: 24 June 2019
We have found the following errors in the article which was recently published in Mathematics [1]:
1. In Example 1, 3 gives rise to the neutrosophic triplet (3, 3, 3). However, 3 has two neutrals: neut(3) = {3, 5}, but 3 does not give rise to a neutrosophic triplet for neut(3)= 5, since anti(3) does not exist in $ℤ 6$ with respect to neut(3) = 5.
2. In Example 2, $ℤ 10$ is not a neutrosophic triplet group. 7 is the classical unitary element of the set $ℤ 10$. Therefore $ℤ 10$ is a neutrosophic extended triplet group.
3. In classical ring theory, for any ring $( R , + , . )$, 0 is the additive identity element. However, in a neutrosophic triplet ring $( N , ∗ , # )$, 0 is an ordinary element and the element 0 is not used in definition. Also N may not have such an element. So, in Definition 8 and subsequent parts of the paper, when using the element 0, the element 0 should be defined.
4. In classical ring theory, for any ring $( R , + , . )$, n∙a is defined by $a + … + a$ and $a n$ is defined by $a … a$ (n times). In neutrosophic triplet ring (NTR), we do not know the definition of $a n$. So before Definition 11, the element $a n$ should be defined.
5. For the proof of Theorem 3, Theorem 1 was used. So Theorem 3 must satisfy the hypothesis of Theorem 1. Also according to definition of $a n$, Theorem 3 should be rewritten.
6. Proposition 1 and its proof are not true. The sentences “if a is not a zero divisor, so a is cancellable” and “if a is cancellable, a is not a zero divisor” are not true. These statements cannot be obtained from the given definitions and theorems.
7. The set P(X) in Example 3 is not neutrosophic triplet field. P(X) has identity elements X and $∅$ for the operations , respectively. Therefore P(X) is a neutrosophic extended triplet group.
8. The counterexamples given for Theorem 5 do not satisfy the distributive law since $1 # ( 1 ∗ 2 ) ≠ ( 1 # 1 ) ∗ ( 1 # 2 )$.
9. In the proof of Theorem 6, the set N is not NTF since $5 # ( 5 ∗ 5 ) ≠ ( 5 # 5 ) ∗ ( 5 # 5 )$.
10. The proof of Theorem 7(2) is not true. If $c ∈ U$, then $f − 1 ( c )$ is a set. If f is not a function, $f − 1 ( c )$ can be equal to an empty set. Then $f − 1 ( c ) ∗ f − 1 ( d )$ is not in $f − 1 ( U )$. We can prove it by the following:
Let $a , b ∈ f − 1 ( U )$. Then $f ( a ) , f ( b ) ∈ U$ and $f ( a ) ⊕ f ( b ) = f ( a ∗ b ) ∈ U$. Hence we get $a ∗ b ∈ f − 1 ( U )$. The proof of $a # b ∈ f − 1 ( U )$ is similar. Also, since $f ( a ) ∈ U$ and $n e u t * ( f ( a ) ) = f ( n e u t * ( a ) ) ∈ U$, we have $n e u t * ( a ) ∈ f − 1 ( U )$. The proof of $n e u t # ( a ) ∈ f − 1 ( U )$ is similar.
11. The proof of Theorem 7(3) is not true. If , then is a set. If f is not a function, can be equal to an empty set. Then $f − 1 ( i ) ∗ f − 1 ( r )$ is not in $f − 1 ( I )$. We can prove it by the following:
Let $a ∈ f − 1 ( U )$ and $r ∈ N T R 1$. Then and $f ( a ) ⊕ f ( r ) = f ( a ∗ r ) ∈ I$. Hence we get $a ∗ r ∈ f − 1 ( I )$. The remaining part of the proof is similar.
12. The proof of Theorem 7(4) should be proven as the following:
Let and $r ∈ N T R 2$. Since f is onto, then $∃ h ∈ J$ exists such that $f ( h ) = j$ and $∃ s ∈ N T R 1$ such that $f ( s ) = r$. Then $h ∗ s ∈ J$ and we get $f ( h ∗ s ) = f ( h ) ⊕ f ( s ) = j ⊕ r ∈ f ( J )$.

## Reference

1. Ali, M.; Smarandache, F.; Khan, M. Study on the Development of Neutrosophic Triplet Ring and Neutrosophic Triplet Field. Mathematics 2018, 6, 46. [Google Scholar] [CrossRef]

## Share and Cite

MDPI and ACS Style

Çeven, Y.; Smarandache, F. Correction: Ali, M., et al. Study on the Development of Neutrosophic Triplet Ring and Neutrosophic Triplet Field. Mathematics 2018, 6, 46. Mathematics 2019, 7, 565. https://doi.org/10.3390/math7060565

AMA Style

Çeven Y, Smarandache F. Correction: Ali, M., et al. Study on the Development of Neutrosophic Triplet Ring and Neutrosophic Triplet Field. Mathematics 2018, 6, 46. Mathematics. 2019; 7(6):565. https://doi.org/10.3390/math7060565

Chicago/Turabian Style

Çeven, Yılmaz, and Florentin Smarandache. 2019. "Correction: Ali, M., et al. Study on the Development of Neutrosophic Triplet Ring and Neutrosophic Triplet Field. Mathematics 2018, 6, 46" Mathematics 7, no. 6: 565. https://doi.org/10.3390/math7060565

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