Isomorphism Of Quotient rings

Question

I have two questions that I need help about.

1) Let $\varphi :R\to S$ be an onto homomorphism, and $I$ ideal in $S.$

Prove that $R/\varphi^{-1} (I)$ isomorphic to $S/I.$

2) Let $I$ be an ideal in $R,$ assume that $I\subset J$, and $J$ ideal in $R.$

Prove that $J/I$ is ideal in $R/I.$

Answer 1

1. Demonstrate, using the definition of ideal, that $\varphi^{-1}(I)$ is an ideal of $R$. Then, you can apply ring's first isomorphism theorem, which gives you the result needed.
2. Let $r \in R, j \in J \Rightarrow r+I \in R/I, j+I \in J/I$. $J$ ideal of $R \Rightarrow rj \in J \Rightarrow rj+I \in J/I \Rightarrow (r+I)(j+I) \in J/I \Rightarrow J/I$ ideal of $R/I$.

P.S.: 2) comes from ring's third isomorphism theorem.

P.S.$^2$: Note that you can reverse the implications in 2).

Answer 2

Hints:

For 1), consider the commutative diagram: \begin{alignat}2 &\begin{gathered}[t]R\\\downarrow\end{gathered}\xrightarrow{\quad\varphi\quad}&\;&\begin{gathered}[t]S\\\downarrow\end{gathered}\\ R/&\varphi^{-1}(I)\xrightarrow{\;\bar\varphi\;}&&S/I \end{alignat} and deduce $\bar\varphi$ is surjective. Note it is injective by construction.

For 2), it is an additive subgroup by the general theorems on quotient groups. You just have to check by hand it is stable by scalar multiplication.