Product of Cosets of Ideals is Subset of Product in Quotient Ring

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

The product $X \circ_\PP Y$ in $\powerset R$ is a subset of their product in $R / J$.

Proof

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Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $22$. New Rings from Old