I am searching for an example of a noncommutative prime ring R and an example of a noncommutative semiprime ring R such that R is not prime, to obtain its maximal right ring of quotients. but I cant find an example until now. please help me...
I am searching for an example of a noncommutative prime ring R and an example of a noncommutative semiprime ring R such that R is not prime, to obtain its maximal right ring of quotients. but I cant find an example until now. please help me...
If you take any prime ring $\displaystyle R$, then any matrix ring over it is also prime. This is because the two sided ideals of $\displaystyle \text{Mat}_n(R)$ are just matrices over two-sided ideals of $\displaystyle R$. I'm pretty sure you can then show that he product of any two-sided ideals is zero, if and only if one is zero. Or, if you are into more advanced solutions, since matrices over a ring are Morita equivalent (their categories of modules are equivalent), and you can characterize primeness via modules, it follows that a matrix ring over a prime ring is prime. So, that's an example of a noncommuative prime ring.
For your second question, have a look here: noncommutative algebra - Semiprime (but not prime) ring whose center is a domain - MathOverflow
Thank you for your help. about my next question, I am not familiar with lie algebra! is there any other example of semiprime ring that is not prime? for example in matrix rings? or if there is not would you explain more about it... please