I have the non-commutative ring R which is
How can I show that every right ideal in the ring R is projective and finitely generated. And how can I use this to show that the right global dimension of R is 1.
you need to be reminded that creating multiple accounts is against this forum's rules! so, as a very mild punishment, you'll only get a partial help on this question!
start with observing that the right ideals of your ring, which i'll call it are exactly in this form: where is any integer and
is any vector subspace of and therefore it's either or isomorphic to also clearly and
which proves that are finitely generated projective submodules of the case is left for you.
hence is a right hereditary ring and therefore every submodule of a free R module has to be projective. so but is not semisimple (why?) and thus
which forces to be 1.
I think you should give me full help, because I dont have any other account on this forum.
What about left ideals? How can I show that there are left ideals in R which are not projective and finitely generated, and how can I conclude that the left global dimension of R is >1.
NB: NonCommAlg has given full help - he was making a joke with that comment. Unfortunately, what you seem to think is full help is getting the complete solution without having to apply any effort. What we think is full help is giving the student who is willing to think a good push in the right direction. Try to be a bit be more grateful - you are clearly in much better shape to answer the question than you were when you first posted it.
Thanks, non-commutative algebra for your help.
I have some questions.
What is e_11 and e_22?
I can see that I_1 and I_2 are finitely generated but how do you get that they are projective?
How do you know that R not semisimple implies that r.gl.dimR is different from 0.
And how can I see that R is not semisimple.
I know that R is right noetherian. Right? Can I use this to say something about that the ring R is not semisimple?
as i mentioned and so it's free and we know every free R-module is projective. also it's in my solution that is a direct summand of and we know that every direct summand of aI can see that I_1 and I_2 are finitely generated but how do you get that they are projective?
projective module is projective.
this is a very basic fact that a ring R is semisimple iff every (right) R-module is projective. thus a ring R is semisimple iff for any R-module M, which is equivalent to say thatHow do you know that R not semisimple implies that r.gl.dimR is different from 0.
because (similary for left global dimension)
recall that the Jacobson radical of a semisimple ring is zero. but in our ring is in the Jacobson radical of R. the reason is that for any the element is easilyAnd how can I see that R is not semisimple.
seen to be invertible in
Note: i won't answer any further questions about this problem anymore. your questions show that you even don't have the minimum background required for problems at this level.
now it's time to get some sleep!