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 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 $$\left({5}\right) \circ_{\mathcal{P}} \left({5}\right) = \left({25}\right)$$.