Factors of quotient rings over some quadratic integer rings

Abstract

Greg Dresden and Wayne M. Dymacek obtained factors of quotient rings over ring of Gaussian integers Z[i]={a+bi/a,b[is an element of a set] Z}. We generalize their idea to obtain factors of quotient rings of the ring of Eisenstein integers Z[omega]={a+b[omega]/a,b[is an element of a set]Z} where [omega]=(-1+[square root]-3)/2 and some other quadratic integer rings Z[omega]={a+b[omega]/a,b[is an element of a set]Z} for [omega]=[square root]d where d is an integer such that d [is equivalent to] 2, 3 (mod 4) or [omega] = (1+[square root]d)/2 where d is an integer such that d [is equivalent to] 1 (mod 4) and in both cases d is squarefree. Moreover, James T. Cross extended naturally the Euler [phi]-function of the ring of integers to the ring of Gaussian integers.We extend it to the ring of Eisenstein integers and some quadratic integers rings.