Ring Operations on Coset Space of Ideal

Theorem
Let $\left({R, +, \circ}\right)$ be a ring.

Let $\mathcal P \left({R}\right)$ be the power set of $R$.

Let $J$ be an ideal of $R$.

Let $X$ and $Y$ be cosets of $J$.

Let $X +_\mathcal P Y$ be the sum of $X$ and $Y$, where $+_\mathcal P$ is the operation induced on $\mathcal P \left({R}\right)$ by $+$.

Similarly, let $X \circ_\mathcal P Y$ be the product of $X$ and $Y$, where $\circ_\mathcal P$ is the operation induced on $\mathcal P \left({R}\right)$ by $\circ$.

Then:
 * The sum $X +_\mathcal P Y$ in $\mathcal P \left({R}\right)$ is also their sum in the quotient ring $R / J$.
 * The product $X \circ_\mathcal P Y$ in $\mathcal P \left({R}\right)$ may be a proper subset of their product in $R / J$.

Proof
As $\left({R, +, \circ}\right)$ is a ring, it follows that $\left({R, +}\right)$ is an abelian group.

Thus by Subgroup of Abelian Group is Normal, all subgroups of $\left({R, +, \circ}\right)$ are normal.

So from the definition of quotient group, it follows directly that $X +_\mathcal P Y$ in $\mathcal P \left({R}\right)$ is also the sum in the quotient ring $R / J$.

The set $\left({5}\right)$ of all integral multiples of $5$ is a principal ideal of the ring $\Z$.

In the ring $\Z / \left({5}\right)$ we have $\left({5}\right) \circ \left({5}\right) = \left({5}\right)$

However, in $\mathcal P \left({\Z}\right)$, we have $\left({5}\right) \circ_\mathcal P \left({5}\right) = \left({25}\right)$.