Definition:Congruence Modulo an Ideal

Let $$\left({R, +, \circ}\right)$$ be a ring, and let $$J$$ be an ideal of $$R$$.

The notation:

$$a \equiv b \left({\bmod\, J}\right)$$

is used to mean:

$$a + \left({-b}\right) \in J$$

Sometimes the notation:

$$a \left({J}\right) b$$

is used.