## Rocky Mountain Journal of Mathematics

### Modules whose certain submodules are essentially embedded in direct summands

#### Abstract

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

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

Dates
First available in Project Euclid: 26 July 2016

https://projecteuclid.org/euclid.rmjm/1469537475

Digital Object Identifier
doi:10.1216/RMJ-2016-46-2-519

Mathematical Reviews number (MathSciNet)
MR3529081

Zentralblatt MATH identifier
06624472

#### Citation

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. https://projecteuclid.org/euclid.rmjm/1469537475

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