Thread: Principal Ideal

Let R be a commutative ring and let I be an ideal in R. If every ideal in R is principal, is the same true for R/I?

I think it's true and am trying to create a contradiction by assuming that J in R/I is not a principal ideal. J must be closed under multiplication by R and addition by itself. So far, I can't get anywhere. A few pointers will be appreciated.

HA, there is a theorem that states every nontrivial proper Ideal in F[x] (ring of polynomials in a field) must be a principal ideal. This is a very important theorem in Field Theory.
But, not necessarily it is true when speaking about any commutative ring with unity.

Originally Posted by Treadstone 71

Let R be a commutative ring and let I be an ideal in R. If every ideal in R is principal, is the same true for R/I?

I think it's true and am trying to create a contradiction by assuming that J in R/I is not a principal ideal. J must be closed under multiplication by R and addition by itself. So far, I can't get anywhere. A few pointers will be appreciated.

From J, construct an ideal J' in R such that J' -> J under R -> R/I. This is a principal ideal by assumption. Use this to show that J is a principal ideal.