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Throughout, is a ring with identity and all modules are left modules. By Jordan-Hölder theorem, all composition series of a module have the same length. So we can now make the following definition. I should remind the reader that, as we showed here, a module has a composition series if and only if it is […]
In this post, is a ring with identity, and all modules are left modules. Here we defined a composition series of a module and we gave some basic examples and results. Here we showed that a module has a composition series if and only if it is both Artinian and Noetherian. Finally, here we discussed […]
As part (1) and part (2) of this post, denotes a ring with identity and all modules are left modules. In the first part of this post, we defined a composition series of a module and gave some basic examples; we also showed that not every module has a composition series. In the second part, […]
Here we defined a PI ring, and also gave some examples and facts about them. In this post, we show that every PI ring satisfying a polynomial of degree also satisfies a multilinear polynomial of degree at most Let’s recall how we prove a Boolean ring is commutative. We say that since for all we […]
I’m not going to repeat what it means for a subset of a ring to be nil or nilpotent, I’ve already done that in this blog several times, see here for example. A theorem of the Israeli algebraist Jacob Levitzki, who was one of Emmy Noether’s students, says that in a (left or right) Noetherian […]
