I am dealing with Commutative Algebra and I found two different (or not?) kinds of quotient of ideals.
One of them, $I/J$, is explained at this topic: How is the quotient of two ideals defined?
The other is the following:
Being $A$ a ring and $I,J$ ideals of $A$, so we define the ideal quotient by $(I:J)=\{a\in A\, :\, aJ\subset I\}.$
Well, one is formed by elements of $A$, other, by classes. However, $(x+I)J\subset I$, so the classes are subsets of $(I:J)$… right?
Does exist any relation between these two quotients?
Many thanks for attention.
Best Answer
Well, $(I:J)$ is the maximal ideal $K$ satisfying $KJ\subseteq I$.
Putting it into the quotient ring $A/I$, we get $$(I:J)/I \cdot (I+J)/I =0$$ (where $0$ is the trivial ideal $I/I$ of $A/I$)
and $(I:J)$ is maximal with this property.