A Special Subring associated with Irreducible Elements in the Ring of integers modulo n

Abstract

For any integer n>1, we determine and count the set of all irreducible elements in
the ring of integers modulo n. A beautiful relation between the cardinality of the
sets of reducible and irreducible elements is shown, for n = p^e . More interestingly, we show that there exist a special subring associated with the set of irreducible elements in the ring of integers modulo n, for n = p^e or n = p^e* m where p is a prime, e > 1 and m is a multiple of distinct primes. Furthermore, for n = p^2 , we show that the subring obtained is a reducible element free and it is always a zero ring. We investigate a number of properties for this special subring. We also discuss how useful this subring could be in other branches of mathematics.