# Math Help - ring

1. ## ring

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

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

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

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

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

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

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

Tonio