Question

Let K be ideal in a ring R . Prove that every ideal in the quotient ring R/K is of the form I/K for some ideal I in R .[Hint: Exercises 19 and 22.]

Short answer

It has been proved thatevery ideal in the quotient ringR/Kis of the formI/K.

Step-by-step solution

Suppose a projection homomorphism

Let J be an ideal in R/K.

Let$\Pi :R\to R/K$ be the projection homomorphism

Show that Π(I)=J

Let I be an ideal in R

then,

$\begin{array}{rcl}I& =& {\Pi }^{-1}\left(J\right)\\ & =& \left\{r\in R|\Pi \left(r\right)\in J\right\}\end{array}$

Furthermore,$\Pi \left(I\right)=J$.

Conclusion

Hence, $\Pi \left(I\right)=\left\{a+K|a\in I\right\}=I/K$

Thus, every ideal in the quotient ring R/Kis of the formI/K.