Thread: 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.

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.

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

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.

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 be a prime number. let and we know that is not projective module. but is a projective

module, because is a field and we know that every module over a field is projective.

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

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.