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Lemma: Factor Rings, Generalization of Congruence Classes

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 \(I\).

Notes

• This is a generalization of the ring of integer congruence classes.
• The factor ring is not to be confused with factorial ring.

Table of Contents

Proofs: 1

1. Lemma: One-to-one Correspondence of Ideals in the Factor Ring and a Commutative Ring

Mentioned in:

Lemmas: 1
Proofs: 2 3
Propositions: 4

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References

Bibliography

1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013