This work provides a comprehensive description of the quotient ring A/mn, where A = Z[ζ] and m = (1 – ζ) is a maximal ideal corresponding to a primitive p-th root of unity ζ. where [1] p is prime number . We explore the structure of A/mn for various values of n, highlighting its properties as a finite Artin local ring [2] and its connection to the p-adic integer ring Zp. By employing techniques from algebraic number theory, we explicitly describe the elements of A/mn, which facilitates the computation of sums and products of any two elements within this ring. This exposition not only elucidates the arithmetic structure of A/mn but also provides valuable insights into its relationship with cyclotomic fields and p-adic analysis [3].