I am currently reading "A book of abstract algebra". I read that ring cosets are defined with respect to an ideal which is the countrepart of normal subgroup in group theory, while group cosets are defined with respect to plain subgroups, not normal subgroups. Why is there such an inhomogeneity in the definition? Thanks, in advance for your time!
Why ring cosets are defined with respect to an ideal while group cosets are defined with respect to a subgroup
abstract-algebragroup-theoryring-theory
Best Answer
Cosets for rings are also defined for subrings, not only for ideals.
Indeed, let $R$ be a ring and $S$ be a subring of $R$. Then $S$ is a subgroup of the additive group of $R$. Since the additive group of $R$ is commutative by the definition of a ring, $S$ is a normal subgroup of the additive group of $R$. Thus she set of cosets $R/S$ becomes a group with respect to the law of addition: $$ (r + S) + (r' + S) = (r + r') + S, $$ where $r + S$ and $r' + S$ are typical cosets belonging to $R/S$. It is clear that with respect to this law of addition, $R/S$ is an abelian group.
However, we can only define a law of multiplication on $R/S$ making $R/S$ a ring when $S$ is an ideal.