Let \((R,+,\cdot)\) be a ring and let \(I\lhd R\) be an ideal. Then there exists a non-empty set \(R/I\) of congruence classes \(r+I~(r\in R)\) and, together with the two new binary operations. * addition: $$(r_1 + I)\oplus (r_2 + I):=(r_1 + r_2) + I,$$ * multiplication: $$(r_1 + I)\odot (r_2 + I):=(r_1 \cdot r_2) + I.$$
it forms a new ring \((R/I,\oplus,\odot)\), called the factor ring (or quotient ring, or residue class ring) with respect to the ideal \(I\).
Proofs: 2 3