Timeline for When are all modules direct factors of a direct product of a fixed one?
Current License: CC BY-SA 3.0
5 events
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| Jul 22, 2011 at 16:42 | comment | added | George C. Modoi | Yes, pure semisimple means pure global dimension zero, i.e. every module is pure projective and pure injective. I found another characterisation of pure semisimplicity in the cited paper by Stovicek, namely $R$ is left pure semisimple iff there is an $R$-module $M$ such that every left $R$-module is a direct summand of a direct sum of copies of $M$. | |
| Jul 20, 2011 at 21:18 | comment | added | David White | Just to clarify, $R$ (left) pure semisimple means it has pure global dimension zero, right? So you know that all flat (left) modules are projective and that every (left) $R$-module is a direct sum of finitely generated $R$-modules, right? I'm just trying to understand the terminology. Is there a better way to think of pure semisimple? | |
| Jun 21, 2011 at 10:20 | history | edited | George C. Modoi | CC BY-SA 3.0 |
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| Jun 21, 2011 at 9:09 | comment | added | George C. Modoi | I forgot: if there exits $M$ such that ${\rm\mathop Mod}(M)={\rm\mathop Prod}(M)$ then it is a test module for projectivity, that is ${\rm\mathop Ext}^1_R(P,M)=0$ iff $P$ is projective. Note that such modules are relatively rare! | |
| Jun 21, 2011 at 8:41 | history | asked | George C. Modoi | CC BY-SA 3.0 |