Definition of quotient ring I have a question about the definition of quotient ring (I mean a commutative ring  case) . 

Let $(a+I)$ be a coset of $R$ and $(b+I)$ be another left coset of $R$.

Then the multiplication in the quotient ring is defined as 

$(a+I)(b+I)=(ab+I)$

However, if we consider the setwise product of $(a+I)$ and $(b+I)$ we know that their product is just a subset of a fixed coset of $I$. Then why should I believe the multiplication in the quotient ring is well-defined?
I suppose you mean that you are considering cosets of an ideal $I$ in a ring $R$, then $$(a+I)(b+I) = ab +aI+bI +I*I = ab + I + I + I= ab + I$$. Because $aI = I$, $I*I = I$ and $I + I=I$ as consequences of the definition of an ideal.
I suppose you mean that you are considering cosets of an ideal $I$ in a ring $R$, then $$(a+I)(b+I) = ab +aI+bI +I*I = ab + I + I + I= ab + I$$. Because $aI = I$, $I*I = I$ and $I + I=I$ as consequences of the definition of an ideal.