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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
Posted in Advanced Algebra
Tagged artinian, composition factors, composition series, length of module, noetherian, simple modules
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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 artinian, composition series, grothendieck group, projective modules
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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 artinian, free modules, left-exact, projective modules, semisimple rings, splitting lemma
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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 artinian, hopkins-levitzki, jacobson radical, matrix rings, nilpotent ideals, noetherian, semisimple rings
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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 algebra, artinian, composition series, length of module, matrix rings, modules, noetherian, unique factorisation
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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 algebra, artinian, jacobson radical, matrix rings, modules, radical of modules, semisimple rings
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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 algebra, artinian, noetherian, semisimple rings, simple modules
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