If and Ideal of a Ring has a prime Ideal, then prime ideal is Ideal of the given Ring [duplicate]
if R is a commutative ring.Let I be an ideal of R and P be a prime ideal of I then show that P is an ideal of R.
let p belongs to P and r belongs to R .then we show that pr,rp belongs to R. if r belongs to P or I then clearly pr,rp belongs to P. therefore we take r in R such that r neither belongs to P nor to I.if some one help me in proving that pr,rp belongs to P.
Let $r\in R$, $p\in P$, $i\in I\setminus P$. Then $i\cdot rp=ir\cdot p\in P$ (because $ir\in I$). Since $P$ is prime and $i\not\in P$, then $rp\in P$.