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I have the following question.

Find a commutative Artinian serial ring $R$ (i.e., a generalized uniserial ring) which is not a principal ideal ring.

Note: A ring $R$ is said to be serial if $R$ is a direct sum of uniserial $R$-modules as right module.

Thank you for your comments to my question.

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  • $\begingroup$ Sorry there was a mistake in my writing: " which is not principal ideal ring. $\endgroup$
    – Najmeh Dehghani
    Commented Sep 4, 2016 at 6:19
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    $\begingroup$ The wiki page en.wikipedia.org/wiki/Serial_module gives you the answer: By a result of Cohen and Kaplansky, a commutative ring RR has the property that its modules are direct sums of cyclic submodules if and only if RR is an Artinian principal ideal ring. By Nakayama, Artinian serial rings have this property on modules. So a commutative Artinian serial ring is a principal ideal ring (i.e., the answer is "no"). $\endgroup$
    – Luc Guyot
    Commented Sep 4, 2016 at 10:14
  • $\begingroup$ Thank you very much for your comment. In fact, I do not think a commutative generalized uniserial ring is Artinian Principal ideal. Beacue by Theorem 4.3 of the paper titled " On the decomosition of modules and generalized left uniserial rings", P. Griffith, we have the following theorem: A commutative ring R is Artinian Principal ideal ring if and only if R is generalized uniserial quasi-Frobenius. " So by this theorem, a generalized uniserial ring which is not quasi-Frobenius is not Atinian Principal ideal ring. Is my argument right? $\endgroup$
    – Najmeh Dehghani
    Commented Sep 4, 2016 at 13:51
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    $\begingroup$ Quoting the wiki page en.wikipedia.org/wiki/Quasi-Frobenius_ring: Commutative Artinian serial rings are all Frobenius rings. You assumed $R$ Artinian in your question. $\endgroup$
    – Luc Guyot
    Commented Sep 4, 2016 at 16:25

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Actually in the commutative case, you don't need deep results.

A commutative ring whose ideals are linearly ordered (a uniserial ring) is obviously Bezout (finitely generated ideals are principal). If it is also Artinian, all ideals are f.g. hence principal, and we are looking at a PIR.

Since a commutative Artinian serial ring factors into a finite product of Artinian uniserial rings, each of which are principal ideal rings, the product is a PIR too.

As you can see, this applies a little more generally to finite products of commutative Noetherian uniserial rings.

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