Quotient ring ( $R / I$ )

For a two-sided ideal

I

R

R / I

has addition and multiplication induced from

R

.

Table of contents
- Congruence modulo an ideal Quotient ring

Congruence modulo an ideal ( $a \equiv b (mod I) ⟺ a - b \in I$ )

Two elements of a commutative ring are congruent modulo an ideal if their difference lies in that ideal, equivalently if their images in the quotient ring agree. This is an equivalence relation compatible with addition and multiplication. For

R = Z_{p} [[T]]

and

I = p^{k} R

, it means every coefficient difference is divisible by

p^{k}

.

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Quotient ring by

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In abstract algebra, a **quotient ring** (or factor ring) is a construction that allows you to create a new ring from a given ring by partitioning it into cosets of a subring. More formally, let \( R \) be a ring and \( I \) be a two-sided ideal of \( R \).

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