# Thread: Ideal in a conmmutative ring

1. ## Ideal in a conmmutative ring

Hi! I have problems with this demostration

Let $I$ be an ideal in a commutative ring $R$. If $J$ is an ideal in $R$ and $I\subseteq{J}$, prove that:

$J/I = \{ r+I : r \in{J} \}$ is an ideal in $R/I$

Demostration

If $J+I\in J/I$, then $J+I\in R/I$ because $J\subset R$, therefore $J/I\subset R/I$.

Is the demonstration correct?

2. ## Re: Ideal in a conmmutative ring

Originally Posted by cristianoceli
Hi! I have problems with this demostration
Let $I$ be an ideal in a commutative ring $R$. If $J$ is an ideal in $R$ and $I\subseteq{J}$, prove that: $J/I = \{ r+I : r \in{J} \}$ is an ideal in $R/I$
Demostration
If $J+I\in J/I$, then $J+I\in R/I$ because $J\subset R$, therefore $J/I\subset R/I$.
Is the demonstration correct?
By Demostration I must assume that you mean PROOF.
You ask: Is the demonstration correct?
That depends upon what is required. What you have posted is correct under certain conditions.
We have no way of knowing what your lecturer requires as a proof.
For me, I expect a student to demonstrate that all conditions for $J/I$ to be an ideal in $R/I$ are meet.
Have you done that?