Zero Divisor Graph for the Ring of Eisenstein Integers Modulo n

Abstract

LetEnbe the ring of Eisenstein integers modulon. In this paper we study the zero divisor graphΓ(En). We find the diameters and girths for such zero divisor graphs and characterizenfor which the graphΓ(En)is complete, complete bipartite, bipartite, regular, Eulerian, Hamiltonian, or chordal.

UNIT AND UNITARY CAYLEY GRAPHS FOR THE RING OF EISENSTEIN INTEGERS MODULO \(n\)

“…, see [4]. This shows that, E n is isomorphic to a direct product of finite local rings (R i , m i ), such that for every i,…”

“…P r o o f. Since q is a prime integer congruent to 1 modulo 3, the ring E q n is the product of the two local rings E/ (a + bω) n and E/ (a + bω) n that have the same number of elements. The ideals a + bω and a + bω are the only maximal ideals of E q n , see [4]. Therefore, by [18, Theorem 3.5], we have diam (G(E…”

“…P r o o f. For each positive integer n, E 3 n is a local ring with the maximal ideal 2 + ω , see [4]. Since ϕ(E 3 n ) = 2 × 3 2n−1 , we have…”

“…The following facts are well known, see for example [4] and [17]. Let ω be a primitive third root of unity.…”

“…Let n be a natural number and let n be the principal ideal generated by n in E. Then the factor ring E/ n is isomorphic to the ring n = {a + bω|a, b ∈ Z n }, where Z n is the ring of integers modulo n. Thus E n is a principal ideal ring. This ring is called the ring of Eisenstein integers modulo n. In [4] this ring is studied and its properties are investigated, its units are characterized and counted. It is easy to see that a + bω is a unit in E n if and only if N (a + bω) is a unit in Z n .…”