Question

Let R be a commutative ring with unity. Prove that the principal ideal generated by x in the polynomial ring R[x] is a prime ideal iff R is an integral domain.

Answer 1

ANSWER :-

Given that R be a commutative ring with unity.

here we need to prove that principal ideal generated by x in the polynomial ring R[x]

and it is a polynomial ring R[x] is a prime ideal

if and only if R is an integral domain.

here R[x] / (x) is also an integral domain

but ker f =(x) and it is a homomorphism function

let x be a prime ideal

and p,q are the functions of prime ideal coefficients

then (p+(x)) (q+(x)

= (0 + x)

= 0 .

R[x] / (x)  is an integral domain.

then the principal ideal generated by x in the polynomial ring R(x) is a prime ideal

and R is an integral domain.

hence proved.