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**Sampras** Let $\displaystyle I $ and $\displaystyle K $ be ideals in a ring $\displaystyle R $ with $\displaystyle J \subseteq I $. Prove that $\displaystyle I/K = \{a+K: a \in I \} $ is an ideal in the quotient ring $\displaystyle R/K $.

Closure under subtraction follows from the fact that $\displaystyle I $ is an ideal.

Suppose $\displaystyle r+K \in R/K $ and $\displaystyle a+K \in I/K $. Then $\displaystyle (r+K)(a+K) = ra+K \in I/K $ since $\displaystyle I $ is an ideal. Likewise, $\displaystyle (a+K)(r+K) = ar+K \in I/K $. So the result follows. Is this correct?