Example

Let R = R, the real numbers, and take d = X² + 1. Then the residue class ring R[X]/(d) is a description of the complex numbers C, with the role of the complex number i being played by X + (d). Indeed,

(X + (d))² = X² + (d) = -1 + (d).

(If you let the complex number a + bi correspond to the class of a + bX, you get the precise correspondence.)

Here, j is the usual embedding of the real numbers into the complex numbers.