Why ring cosets are defined with respect to an ideal while group cosets are defined with respect to a subgroup? 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!
 A: 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.
A: 
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.

Those details really do not matter to cosets. I would put it this way:
The only thing you need for cosets is a group and a subgroup.
The cosets of an ideal or even of a subring are completely determined by the underlying abelian group of the ideal or subring, and they do not really care whether or not the subgroup has extra properties.
Now, why do extra things like normal subgroups and ideals important?
They let us do extra stuff with the cosets: cosets made using a normal subgroup will have their own group structure. cosets made in a ring using an ideal will have a ring structure.
