Every ideal of S is principal when f:R⇒S be a surjective homomorphism of rings with identity.
Given that,
Let f:R⇒S be a surjective homomorphism of rings with identity.
We have to find if R is a PID, prove that every ideal in S is principal.
We know that,
Let I be the ideal of S
Since f is sufficient homomorphism.
So, f⁻¹(I) is an ideal of R.
Since R is PID so ∈ r∈R such that
f⁻¹(I) = <r>
I = <f(r)>
Therefore, Every ideal of S is principal when f:R⇒S be a surjective homomorphism of rings with identity.
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