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Sajana, S., Bharathi, D. (2019). Number theoretic properties of the commutative ring Zn. International Journal of Research in Industrial Engineering, 8(1), 77-88. doi: 10.22105/riej.2019.159539.1065
Sh. Sajana; D. Bharathi. "Number theoretic properties of the commutative ring Zn". International Journal of Research in Industrial Engineering, 8, 1, 2019, 77-88. doi: 10.22105/riej.2019.159539.1065
Sajana, S., Bharathi, D. (2019). 'Number theoretic properties of the commutative ring Zn', International Journal of Research in Industrial Engineering, 8(1), pp. 77-88. doi: 10.22105/riej.2019.159539.1065
Sajana, S., Bharathi, D. Number theoretic properties of the commutative ring Zn. International Journal of Research in Industrial Engineering, 2019; 8(1): 77-88. doi: 10.22105/riej.2019.159539.1065

Number theoretic properties of the commutative ring Zn

Article 7, Volume 8, Issue 1, Spring 2019, Page 77-88  XML PDF (466.21 K)
Document Type: Review Paper
DOI: 10.22105/riej.2019.159539.1065
Authors
Sh. Sajana email 1; D. Bharathi2
1Department of Mathematics and Statistics, P.R. Govt. College (A), Kakinada, Andhra Pradesh, India.
2Department of Mathematics, S.V. University, Tirupati, Andhra Pradesh, India.
Abstract
This paper deals with the number theoretic properties of non-unit elements of the ring Zn. Let D be the set of all non-trivial divisors of a positive integer n. Let D1 and D2 be the subsets of D having the least common multiple which are incongruent to zero modulo n with every other element of D and congruent to zero modulo n with at least one another element of D, respectively. Then D can be written as the disjoint union of D1 and D2 in Zn. We explore the results on these sets based on all the characterizations of n. We obtain a formula for enumerating the cardinality of the set of all non-unit elements in Zn whose principal ideals are equal. Further, we present an algorithm for enumerating these sets of all non-unit elements.
Keywords
Non-unit elements; Non-trivial divisor; Least common multiple; Congruent; Finite commutative ring; Principal ideal
Main Subjects
Other
References

[1]     Nathanson, M. B. (1980). Connected components of arithmetic graphs. Monatshefte für mathematik, 89(3), 219-222.

[2]     Apostol, T. M. (1989). Introduction to analytic number theory. Springer International Student Edition.

[3]     Beck, I. (1988). Coloring of commutative rings. Journal of algebra, 116, 208-226.

[4]     Anderson, D. F., & Livingston, P. S. (1999). The zero-divisor graph of a commutative ring. Journal of algebra, 217(2), 434-447.

[5]     Bhattacharya, P. B., Jain, S. K., & Nagpaul, S. R. (1994). Basic abstract algebra. Cambridge University Press.

[6]     Hungerford, T. W. (2012). Abstract algebra: an introduction. Cengage Learning.

[7]     Nathanson, M. B. (2008). Elementary methods in number theory (Vol. 195). Springer Science & Business Media.

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