Rocky Mountain Journal of Mathematics

Modules whose certain submodules are essentially embedded in direct summands

Yeliz Kara and Adnan Tercan

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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.

Article information

Rocky Mountain J. Math., Volume 46, Number 2 (2016), 519-532.

First available in Project Euclid: 26 July 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 16D70: Structure and classification (except as in 16Gxx), direct sum decomposition, cancellation
Secondary: 16D50: Injective modules, self-injective rings [See also 16L60] 16D80: Other classes of modules and ideals [See also 16G50]

Extending module exchange property $ C_12 $-module tangent bundle


Kara, Yeliz; Tercan, Adnan. Modules whose certain submodules are essentially embedded in direct summands. Rocky Mountain J. Math. 46 (2016), no. 2, 519--532. doi:10.1216/RMJ-2016-46-2-519.

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