5
$\begingroup$

I'm looking for some examples of problems, possibly hard problems, which were solved with the use of primary decomposition. I am also looking for examples of its uses and where it provided further insights into the structure of rings. More detail if necessary.

$\endgroup$

closed as too broad by user26857, Namaste, Claude Leibovici, mlc, kingW3 Jun 2 '17 at 13:15

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ You would do well to begin with the Wikipedia article on Primary decomposition. Note the curious combination of names in the Lasker-Noether theorem. Emanuel Lasker is now mainly remembered for his domination of championship chess, but he was accomplished as a research mathematician as well as at games other than chess. $\endgroup$ – hardmath Jun 7 '17 at 2:34
  • $\begingroup$ "More detail if necessary." Yes, that would be helpful. $\endgroup$ – hardmath Jun 7 '17 at 13:12
  • 1
    $\begingroup$ I realize this question is quite general, so may be hard to answer -- thus the close votes. Maybe the following will motivate the question: I realized that primary decomposition can be dualized, so that every submodule of an Artinian module has a reduced "coprimary" decomposition. The uniqueness theorems also dualize. But I was trying to figure out what this could be useful for, and one place to start would be analogizing the places where primary decomposition is used. $\endgroup$ – Dean Young Jun 7 '17 at 16:50