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 $X +_\PP Y$ be the sum of $X$ and $Y$, where $+_\PP$ is the operation induced on $\powerset R$ by $+$.

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