Ring Operations on Coset Space of Ideal

Theorem
Let $\struct {R, +, \circ}$ be a ring.

Let $\powerset R$ 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 $\powerset R$ by $+$.

Similarly, let $X \circ_\mathcal P Y$ be the product of $X$ and $Y$, where $\circ_\mathcal P$ is the operation induced on $\powerset R$ by $\circ$.

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

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

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

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

The set $\ideal 5$ of all integral multiples of $5$ is a principal ideal of the ring $\Z$.

In the ring $\Z / \ideal 5$ we have:
 * $\ideal 5 \circ \ideal 5 = \ideal 5$

However, in $\powerset \Z$, we have $\ideal 5 \circ_\mathcal P \ideal 5 = \ideal {25}$.