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Quotient module

Definition

Given an ideal of a ring $I \lhd R$, the quotient ring $R/I$ is the set of cosets $r + I = \{i \in N: r + i\}; r \in R$, with addition defined by $(r_1 + I) + (r_2 + I) = (r_1 + r_2) + I$, and scaling defined by $(r_1+N).(r_2+N)=r_1.r_2+N$.

Properties

$R/I$ is an integral domain if and only if $I$ is a prime ideal.

$R/I$ is a field if and only if $I$ is a maximal ideal.

all these pages adapted with probably insufficient credit from my university's lecture notes