# Thread: Ideal in a conmmutative ring

1. ## Ideal in a conmmutative ring

Hi! I have problems with this demostration

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

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

Demostration

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

Is the demonstration correct?

2. ## Re: Ideal in a conmmutative ring

Originally Posted by cristianoceli
Hi! I have problems with this demostration
Let $\displaystyle I$ be an ideal in a commutative ring $\displaystyle R$. If $\displaystyle J$ is an ideal in $\displaystyle R$ and $\displaystyle I\subseteq{J}$, prove that: $\displaystyle J/I = \{ r+I : r \in{J} \}$ is an ideal in $\displaystyle R/I$
Demostration
If $\displaystyle J+I\in J/I$, then $\displaystyle J+I\in R/I$ because $\displaystyle J\subset R$, therefore $\displaystyle J/I\subset R/I$.
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?