# Tag Archives: artinian

## Commutative Algebra 38

Artinian Rings The main result we wish to prove is the following. Theorem. A ring A is artinian if and only if it is noetherian and , where denotes the Krull dimension. Note Recall that means all prime ideals of A … Continue reading

## Commutative Algebra 37

Artinian Modules Instead of the ascending chain condition, we can take its reverse. Definition. Let M be an A-module. Consider the set of submodules of M, ordered by inclusion, i.e. if and only if . We say M is artinian … Continue reading

## 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

## 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 | | 2 Comments

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

## 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