Open Access
2016 Modules whose certain submodules are essentially embedded in direct summands
Yeliz Kara, Adnan Tercan
Rocky Mountain J. Math. 46(2): 519-532 (2016). DOI: 10.1216/RMJ-2016-46-2-519


It is well known that, if the ring has acc on essential right ideals, then for every quasi-continuous module over the ring, the finite exchange property implies the full exchange property. In this paper, we obtain the former implication for the generalizations of quasi-continuous modules over a ring with acc on right annhilators of elements of the module. Moreover, we focus on direct sums and direct summands of weak $C_{12} $ modules i.e., modules with the property that every semisimple submodule can be essentially embedded in a direct summand. To this end, we prove that since weak $ C_{12} $ is closed under direct sums. Amongst other results, we provide several counterexamples including the tangent bundle of a real sphere of odd dimension over its coordinate ring for the open problem of whether weak $ C_{12}$ implies the $ C_{12} $ condition.


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Yeliz Kara. Adnan Tercan. "Modules whose certain submodules are essentially embedded in direct summands." Rocky Mountain J. Math. 46 (2) 519 - 532, 2016.


Published: 2016
First available in Project Euclid: 26 July 2016

zbMATH: 06624472
MathSciNet: MR3529081
Digital Object Identifier: 10.1216/RMJ-2016-46-2-519

Primary: 16D70
Secondary: 16D50 , 16D80

Keywords: $ C_12 $-module , Exchange property , Extending module , Tangent bundle

Rights: Copyright © 2016 Rocky Mountain Mathematics Consortium

Vol.46 • No. 2 • 2016
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