Commutative Algebra/Fractions, annihilator, quotient ideals

The quotient of two ideals

We note some properties:

The points 1. and 2. make calling those ideals "quotient" plausible, 3. and 4. less so (although the ideal still gets smaller when adding something to the denominator or shrinking the numerator).

Proof:

1. $I\cdot J\subseteq K\iff \mathrm{\forall }i\in I:iJ\subseteq K\iff I\subseteq (K:J)$

2. $J\subseteq I\iff \mathrm{\forall }r\in R:rJ\subseteq J\subseteq I$

3.

$\begin{array}{cc}r\in (I:J+K)& \iff r(J+K)\subseteq I\\ & \iff rJ\subseteq I\wedge rK\subseteq I\\ & \iff r\in (I:J)\cap (I:K),\end{array}$

where the middle equivalence follows since $r(J+K)$ is the smallest ideal containing $rJ$ and $rK$, and thus is contained in every ideal where the latter two are contained.

4.

$\begin{array}{cc}r\in ({\cap }_{i\in I}{I}_i:K)& \iff rK\in {\cap }_{i\in I}{I}_i\\ & \iff \mathrm{\forall }i\in I:rK\in I_i\\ & \iff \mathrm{\forall }i\in I:r\in (I_i:K)\\ & \iff r\in {\cap }_{i\in I}(I_i:K)\end{array}$

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Exercises

• Exercise 19.1.1: Prove that for a ring $R$ and any ideal $I\le R$, $(R:I)=R$ and $(I:R)=I$.