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

Sources

• 1969: M.F. Atiyah and I.G. MacDonald: Introduction to Commutative Algebra: Chapter $1$: Rings and Ideals