why coset of $I$ is a set of the form $a + I =\{a+s|s\in I\}?$ why not $a.I$? Definition of Coset in Group theory : let $H  \subset G$.Suppose  if we multiply all element of $H$ on the  left/right by one fixed element of $G$ ,then set of product  is called coset
Definition of Coset Ring theory: let $I$ is  an  ideal  of $R$ then a  coset of $I$  is a  set of the form $a + I =\{a+s|s\in I\}$
My confusion:  why coset of $I$  is a  set of the form $a + I =\{a+s|s\in I\}?$ why not $a.I$?
My thinking  : If we replaced $H$  with ideal $I$ and   $G$  with $R$   then  coset of $I$ will be  a  set of the form $a . I =\{a.s|s\in I\}$
 A: A group is defined to be a set $G$ together with a binary operation $*:G \times G \to G$ satisfying certain conditions. A (left) coset of a subgroup $H$ of $G$ is a set of the form $g*H = \{ g*h : h \in H \}$ for a fixed element $g \in G$.
So, formally, it is not correct to that you get a coset by "multiplying" an element of the group by elements of the subgroup. The definition depends on the group operation. Sometimes we refer to the operation as "multiplication" and we may decide to write $gh$ as an abbreviation for $g*h$, but that is just a notational convenience. At other times it is more convenient to use $+$ as the operation, such as for the integers under addition, and in that case we would denote the coset by $g+H$ rather than $g*H$.
In a ring, we have two binary operations, which are conventionally referred to as addition and multiplication, although again that is just fo0r convenience. The elements of a ring form a group under addition, but except when the ring consists of $0$ only, they do not form a group under multiplication. So the whole idea of a coset under multiplication doesn't make sense.
