Final answer:
To prove that every ideal of the quotient ring R/I is principal, we can start by defining the quotient ring. Let J be any ideal in the quotient ring R/I. We want to show that J is principal, which means it can be generated by a single element.
Step-by-step explanation:
To prove that every ideal of the quotient ring R/I is principal, we can start by defining the quotient ring. The quotient ring R/I consists of equivalence classes of elements of R, where two elements a and b in R are considered equivalent if their difference a - b is in the ideal I. Now, let J be any ideal in the quotient ring R/I. We want to show that J is principal, which means it can be generated by a single element.
Since J is an ideal in R/I, it is closed under addition and multiplication by elements of R/I. Let's consider the set A of representatives of the elements in J. We can show that A is an ideal in R that contains I. Since R is a principal ideal domain, A is principal and can be generated by a single element, say a. Therefore, J can also be generated by the equivalence class [a] in R/I, making J a principal ideal.