ring

**Sampras** · October 27th 2009, 01:49 PM

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?

**tonio** · October 27th 2009, 02:01 PM

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

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