Morita invariance, I - Githubissues

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Data for the Database of Ring theory

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Morita invariance, I #59

Closed dyunov closed 8 months ago

dyunov commented 8 months ago

The following Morita invariant properties from your edits on Wikipedia are not marked as such in DaRT:

[x] simple
[x] semihereditary
[x] hereditary
[x] nonsingular
[x] coherent
[x] semiprimary
[x] perfect
[x] semiperfect
[x] semilocal

Specialization of $R{11}$ and $R{12}$ would help with negative metaproperties.

rschwiebert commented 8 months ago

~~The ones I don't have negative examples/relations for:~~

[ ] Frobenius last Remark in this article seems to give an example. Kerner, Otto, and Kunio Yamagata. "Morita algebras." Journal of Algebra 382 (2013): 185-202.

[ ] Goldie

[ ] IBN (corollary 2.9 of this but i hope to find something simpler.

~~[x] reversible~~

Taken care of below.

dyunov commented 8 months ago

(links found on Lam II p482)

[x] Goldie and ACC annihilator do not hold in $M_2(R)$ where $R$ is commutative Goldie from J. Kerr "An example of a Goldie ring whose matrix ring is not Goldie".
[x] IBN not Morita invariant is Ex18.11 p501 ("using the technique of coproducts from Bergman [74], it can be shown"...) It passes to matrix rings though, see Ex1.14 p18.
[x] Frobenius: Ex18.7B(3) p502.
[x] Finite uniform dimension is Morita invariant, Ex18.6 p501.