Quotient Groups versus Quotient Rings I have a doubt in the definition of Quotient Rings Take $R$ to be a ring and $I$ to be an ideal in it.Let us label all element of $I$ be $a_{i}$
The Quotient Rings are defined as $\frac{Q}{I} = {x+I} $ where ${x+I}$ are the set of all cosets. Now the multiplication and addition are defined usually $(x+I)+(y+I)=(x+y)+I$ and $(x+I)*(y+I)=(x*y)+I$
Now i have observed a special property in addition of cosets $(x+I)+(y+I)=(x+y)+I$ here what i mean it is just not defined by addition but what actually holds $(x+I)+(y+I)=x+a_{1}+y+a_{2}=x+y+a_{1}+a_{2} \subset
 (x+y)+I$ This will hold since the group defined is abelian
and for the other direction $\forall a_{i} (x+y)+a_{i}=x+0+y+a_{i}\subset (x+I)+(y+I)$ . I have proved this for ideals but it can be proved in the same idea for Quotient groups. However this just doesn't hold for multiplication for Quotient rings i mean I am not able to prove the other direction 
$ \forall i \  x*y +a_{i} = (x+a_{j})*(x+a_{k}) $ . I am fairly confident that this itself won't hold, can anyone give me an example for this.
What i think this means that addition of cosets is obvious in the sense you add all pair of elements in the coset and you get a new coset, which will be defined as the addition of cosets, however not for multiplication of cosets. My question is what is the reasoning behind this definition? Why not choose multiplication of cosets to be multiplication of each pair of elemnts and forming a new coset. Granted it will not be an existing coset but i think we can choose something stronger than an ideal for this new definition 

Multiplication of cosets works as such; for $i,j \in I$ and $a,b \in R$ we have that $a+i \in a + I$ and $b+j \in b+J$, so choosing $a+i$ and $b+j$ as our coset representatives, consider the multiplication $(a+i)(b+j)$. 
This is of course $ab+ib+aj+ij$, but since $I$ is an ideal, $ib, aj, ij$ are all in $I$. So then the coset $ab+I$ contains $ab+ib+aj+ij$, but we may choose any $I$ element as our representative.
