Dfference between strongly prime and prime ideal An ideal $P\subset R$ is strongly prime, if for any $x$ and $y$ in the quotient field of $R$, $xy\in P$ implies $x\in P$ or $y\in P$. 
What is the difference between strongly prime ideal of $R$ and a prime ideal of $R$? 
It seems that they are same .Thank you for your helping ..
 A: As should be abundantly clear from the comments yesterday, they are far from similar. Jorst pointed out that $\frac23\cdot 3\in (2)\subseteq\Bbb Z$, and yet neither one of $\frac23$ or $3$ lies in $(2)$. I'll answer talking about a domain $R$ with field of quotients $Q$.
As for your question "what is the difference?" the first thing to point out is that the definitions quantify over different sets. "For all $x,y\in R$" is a wholly different statement from "For all $x,y\in Q$". 
By adapting the argument above, you can show that no nonzero ideal of $\Bbb Z$ is strongly prime. For any $(p)$ with $p$ a nonzero prime, you can just take a prime $q$ other than $p$ and note $q\cdot frac{p}{q}\in (p)$, but $\frac{p}{q}, q\notin (p)$. The zero ideal, however, is always strongly prime.
What can be said? Of course, any strongly prime ideal not equal to $R$ is also a(n ordinary) prime ideal. I'm not sure how many sources include/exclude $R$ from this definition you gave.
Secondly, if $R$ is allowed into the definition above, then if $R$ is strongly prime, then the ideals of $R$ are linearly ordered. (This happens because $xx^{-1}=1\in R$ implies $x$ or $x^{-1}$ is in $R$, and this is a way to define a valuation domain.)
