1. ring

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?

2. Originally Posted by 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?

Yes, it seems it is. But that J above must be K...right?

Tonio