Tag Archives: artinian

Projective Modules and the Grothendieck Group

This is a continuation of the previous article. Throughout this article, R is an artinian ring (and hence noetherian) and all modules are finitely-generated. Let K(R) be the Grothendieck group of all finitely-generated R-modules; K(R) is the free abelian group generated by [M] for simple … Continue reading

Posted in Notes | Tagged , , , | Leave a comment

Projective Modules and Artinian Rings

Projective Modules Recall that Hom(M, -) is left-exact: for any module M and exact , we get an exact sequence Definition. A module M is projective if Hom(M, -) is exact, i.e. if for any surjective N→N”, the resulting HomR(M, N) → HomR(M, N”) is … Continue reading

Posted in Notes | Tagged , , , , , | 2 Comments

Jacabson Radical

Recall that the radical of the base ring R is called its Jacobson radical and denoted by J(R); this is a two-sided ideal of R. Earlier, we had proven that a ring R is semisimple if and only if it is artinian and J(R) = … Continue reading

Posted in Notes | Tagged , , , , , , | Leave a comment

Composition Series

Positive integers can be uniquely factored as a product of primes. Here, we would like to prove a counterpart for modules. Now there are two ways to “factor” a module M; a more liberal way takes a submodule N which gives us composition … Continue reading

Posted in Notes | Tagged , , , , , , , | Leave a comment

Radical of Module

As mentioned in the previous article, we will now describe the “bad elements” in a ring R which stops it from being semisimple. Consider the following ring: Since R is finite-dimensional over the reals, it is both artinian and noetherian. However, R is not … Continue reading

Posted in Notes | Tagged , , , , , , | Leave a comment

Noetherian and Artinian Rings and Modules

We saw the case of the semisimple ring R, which is a (direct) sum of its simple left ideals. Such a ring turned out to be nothing more than a finite product of matrix algebras. One asks if there is a … Continue reading

Posted in Notes | Tagged , , , , | Leave a comment