Definition:Ideal Quotient of Commutative Ring

Definition
Let $R$ be a commutative ring with unity.

Let $\mathfrak a, \mathfrak b$ be ideals of $R$.

The ideal quotient of $\mathfrak a$ and $\mathfrak b$ is the ideal defined as:
 * $\ds \ideal {\mathfrak a : \mathfrak b} := \set {x \in R : x \mathfrak b \subseteq \mathfrak a}$

where:
 * $x \mathfrak b := \set {x y : y \in \mathfrak b}$

Also known as
If $\mathfrak b$ is a principal ideal $\ideal x$, we write:
 * $ \ideal {\mathfrak a : x} := \ideal {\mathfrak a : {\ideal x} }$

Also see

 * Ideal Quotient is Ideal