Number theoretic properties of the commutative ring Zn

Document Type: Review Paper


1 Department of Mathematics and Statistics, P.R. Govt. College (A), Kakinada, Andhra Pradesh, India.

2 Department of Mathematics, S.V. University, Tirupati, Andhra Pradesh, India.


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.


Main Subjects

[1]     Nathanson, M. B. (1980). Connected components of arithmetic graphs. Monatshefte für mathematik89(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 algebra217(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.