On Symmetric Additive Mappings and Their Applications
Abstract
:1. Introduction
2. The Results
- (i)
- (ii)
- .
- (i)
- (ii)
- is a commutative integral domain.
- (i)
- (ii)
- is a commutative integral domain.
- (i)
- (ii)
- is a commutative integral domain.
- (i)
- ∀
- (ii)
- ∀
- (iii)
- ∀
- 1.
- 2.
- is a commutative integral domain.
- (i)
- Assume thatTaking into account Lemma 1 and the torsion restriction, we find that
- (ii)
- Proceeding in the same way as in , we conclude.
- (iii)
- It is given that
- (i)
- (ii)
- is a commutative integral domain.
- (i)
- (ii)
- (i)
- Given thatUsing Lemma 1 and torsion restriction, we have
- (ii)
- It follows from the first implication with a slight modification.
- (i)
- (ii)
- (i)
- (ii)
- (i)
- Let us assume that
- (ii)
- Similarly, if for all , then the same reasoning proves the required result.
- (i)
- (ii)
- (i)
- Replacing with for , in the given condition, we obtain
- (i)
- (ii)
- (i)
- Given thatMaking use of Lemma 1 and torsion restriction, we see that
- (ii)
- It follows from the first implication with a slight modification.
- (i)
- (ii)
- (i)
- (ii)
- (i)
- We assume thatMaking use of Lemma 1, we see that
- (ii)
- Proceeding in the same way as in (i), we conclude.
- (i)
- (ii)
3. Applications
- 1.
- 2.
- is commutative.
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Ali, S.; Alsuraiheed, T.; Varshney, V.; Wijayanti, I.E. On Symmetric Additive Mappings and Their Applications. Symmetry 2023, 15, 1874. https://doi.org/10.3390/sym15101874
Ali S, Alsuraiheed T, Varshney V, Wijayanti IE. On Symmetric Additive Mappings and Their Applications. Symmetry. 2023; 15(10):1874. https://doi.org/10.3390/sym15101874
Chicago/Turabian StyleAli, Shakir, Turki Alsuraiheed, Vaishali Varshney, and Indah Emilia Wijayanti. 2023. "On Symmetric Additive Mappings and Their Applications" Symmetry 15, no. 10: 1874. https://doi.org/10.3390/sym15101874
APA StyleAli, S., Alsuraiheed, T., Varshney, V., & Wijayanti, I. E. (2023). On Symmetric Additive Mappings and Their Applications. Symmetry, 15(10), 1874. https://doi.org/10.3390/sym15101874