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Describe rings whose left modules are free-like, projective-like or injective-like

All our rings are associative rings with identity and modules are left (right) unital. An R-module Q is called free-like if it is a direct sum of cyclic modules. Also, an R-module P is called projective-like (or cyclically-pure-projective) if it is a direct summand of a free-like module. The notion of injective-like (or cyclically-pure-injective) module is a dual notion of projective-like module. Commutative rings for which every module is free-like are characterized by K¨othe, Cohen and Kaplansky.

Theorem 1. (K¨othe, [Math. Z. 39 (1935), 31-44]). Over an Artinian principal ideal ring, each module is a direct sum of cyclic modules. Furthermore, if a commutative Artinian ring has the property that allits modules are direct sums of cyclic modules, then it is necessarily a principal ideal ring.

Theorem 2. (Cohen and Kaplansky, [Math. Z. 54 (1951), 97-101]). If R is a commutative ring such that each R-module is a direct sum of cyclic modules then R must be an Artinian principal ideal ring.

The corresponding problem in the noncommutative case remains open. Nakayama [Proc. Imp. Acad. Tokyo, vol. 16 (1940), 285-289] gave example of a noncommutative right Artinian ring R whose each right module is a direct sum of cyclic modules but R is not a principal right ideal ring.

The purpose of this research is to provide answers for the following  problems.

(1) (K¨othe) Describe rings over which each left (left and right) module is a direct sum of cyclic modules.

(2). Describe rings whose ideals are direct sums of cyclics

(3)  Describe rings whose prime ideals are direct sums of cyclics

(4)..........

Related Papers:

1-M. Behboodi (Joint with S.H. Shojaee)
Commutative local rings whose ideals are direct sum of cyclic modules
Algebr. Represent Theor. (2013), DOI: 10.1007/s10468-013-9427-x    

2-M. Behboodi (Joint with A. Ghorbani and A. Moradzadeh-Dehkordi)
Commutative Noetherian local rings whose ideals are direct sums of cyclic modules
J. Algebra 345 (2011), 257-265 

3- M. Behboodi (Joint with A. Ghorbani, A. Moradzadeh-Dehkordi, and S. H. Shojaee)
On left Kothe rings and an analogue of the Cohen-Kaplansky theorem
Proc. Amer. Math. Soc. (Accepted)

4- M. Behboodi (Joint with A. Moadzadeh-Dehkordi)
On Commutative rings whose prime ideals are direct sum of cyclics
Arch. Math. (Brno) 48 (2012), 291-299

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Non Resident Researcher (non-resident), School of Mathematics, INSTITUTE  FOR  RESEARCH  IN  FUNDAMENTAL SCIENCES, 2010-2011