1. ## non-Commutative algebra

I is a two sided ideal in a ring R AndM is left R-module.M/IM becomes a module over the quotient ring R/I.
How can I show that if M/IM is a projective R/I-module then M is a projective R-module.

2. Originally Posted by peteryellow
I is a two sided ideal in a ring R and M is left R-module. M/IM becomes a module over the quotient ring R/I.
How can I show that if M/IM is a projective R/I-module then M is a projective R-module.
you can't show that because it's false! can you find a counter-example? maybe the question is this: if M is a projective R-module, then M/IM is a projective R/I-module.

3. No I have no idea if this is false. This is my question.

4. See my question is.

Let I be a two-sided ideal in a ring R and let M be a left R-module. IM contained in M is a submodule generated by all products xm where x is in I and m is in M. The quotient M/IM becomes a module over the quotient ring R/I.

Suppose that I is nilpotent, i.e., taht I^N =0 for some N>=1 and suppose that M is finitely generated flat R-module. Show that if M/IM is a projective R/I-module then M is a projective R-module.

Sorry I didnot wrote the whole question.

5. Originally Posted by peteryellow
No I have no idea if this is false. This is my question.
it's obviously false! a simple counter-example is this: let $\displaystyle p$ be a prime number. let $\displaystyle R=\mathbb{Z}, \ I=p\mathbb{Z},$ and $\displaystyle M=\mathbb{Q}.$ we know that $\displaystyle \mathbb{Q}$ is not projective $\displaystyle \mathbb{Z}-$module. but $\displaystyle M/IM$ is a projective $\displaystyle R/I$

module, because $\displaystyle R/I=\mathbb{Z}/p$ is a field and we know that every module over a field is projective.

6. But what about my full question do you also think that this is also wrong?

7. Originally Posted by peteryellow
See my question is.

Let I be a two-sided ideal in a ring R and let M be a left R-module. IM contained in M is a submodule generated by all products xm where x is in I and m is in M. The quotient M/IM becomes a module over the quotient ring R/I.

Suppose that I is nilpotent, i.e., taht I^N =0 for some N>=1 and suppose that M is finitely generated flat R-module. Show that if M/IM is a projective R/I-module then M is a projective R-module.

Sorry I didnot wrote the whole question.
ok, well, i don't know how you forgot to mention strong conditions like these!! i'll think about this "new" problem and i'll get back to you when i have a solution.