Section 26 Homomorphisms and Factor Rings

26.10 Definition Ideal

A First Course in Abstract Algebra, p.241

An additive subgroup $N$ of a ring $R$ satisfying the properties $aN\subseteq N$ and $Nb\subseteq N$ (for any element in $N$) for all $a,b\in R$ is an ideal.

In Ring homomorphism, $Ker(\varphi )$ is the ideal.

• It’s similar to normal subgroup in group theory. see 13.19 Definition normal
• In addition to the criterium in normal subgroups, elements in an ideal must also satisfy that: for each elements in the ring, the multiplication between the element and an element in the ideal must be still in the ideal.