RINGS WHOSE CYCLIC MODULES ARE DIRECT SUMS OF EXTENDING MODULES

AYDOĞDU, PINAR ; ER, NOYAN ; ERTAŞ, NİL ORHAN

Glasgow mathematical journal, Tome 54 (2012) no. 3, pp. 605-617

Résumé

Dedekind domains, Artinian serial rings and right uniserial rings share the following property: Every cyclic right module is a direct sum of uniform modules. We first prove the following improvement of the well-known Osofsky-Smith theorem: A cyclic module with every cyclic subfactor a direct sum of extending modules has finite Goldie dimension. So, rings with the above-mentioned property are precisely rings of the title. Furthermore, a ring R is right q.f.d. (cyclics with finite Goldie dimension) if proper cyclic (≇ RR) right R-modules are direct sums of extending modules. R is right serial with all prime ideals maximal and ∩n ∈ NJn = Jm for some m ∈ N if cyclic right R-modules are direct sums of quasi-injective modules. A right non-singular ring with the latter property is right Artinian. Thus, hereditary Artinian serial rings are precisely one-sided non-singular rings whose right and left cyclic modules are direct sums of quasi-injectives.

Détail

DOI : 10.1017/S0017089512000183

Mots-clés : Primary 16D10, 16D70, 16P20, 16P40

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