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FC-Pure Injectivity and FC-Pure Flatness and Its Relationship to Two Famous Problems of Kothe and Kaplansky

Let S be a class of left R-modules.  An R-module M is said to be S-pure injective (resp. S-pure projective) if M has the injective (resp. projective) property respect to each S-pure exact sequence. Also, an R-module M is said to be S-pure flat if M has the flat property respect to each S-pure exact sequence. As  Puninski (1994), let us fix a few of the most common purities S of R-modules.

1. Purity. If S = { the class of all finitely presented left modules over R}, then we have the left Cohn’s purity. In this case, S-pure exact, S-pure projective, S-pure injective and S-pure flat are commonly called pure exact, pure projective, pure injective and pure flat, respectively
2. RD-purity. If S = {R/rR | r \in R} (R/rR is called cyclically presented), then we have the RD-purity. In this case, S-pure exact, S-pure projective, S-pure injective and S-pure flat are commonly called RD-exact (or relative divisible exact), RD-projective, RD-injective and RD-flat, respectively.
3. I-purity (or cyclically purity) . If S = {R^n/K | K is a cyclic submodule of the R-module R^n}, then we have I-purity or cyclically purity. In this case, S-pure exact, S-pure projective, S-pure injective and S-pure flat are commonly called I-pure exact, I-pure projective, I-pure injective and I-pure flat, respectively.
4. C-purity. If S = {R/I | I is a left ideal of R}, then we have the cyclically purity (or c-purity). In this case, S-pure exact, S-pure projective, S-pure injective and S-pure flat are commonly called C-pure exact, C-pure projective, C-pure pure injective and C-pure flat, respectively.
5. FC-purity. Let S = {R/I | I is a finitely generated left ideal of R}. In this case, since each element R/I of S is finitely presented cyclic R-module, we have the FC-purity (or finitely presented cyclically purity). Note that Puninski, use the phrase Warfield purity to mean FC-purity. In this case, S-pure exact, S-pure projective, S-pure injective and S-pure flat are commonly called FC-pure exact, FC-pure projective, FC-pure injective and FC-pure flat, respectively.

The question of which commutative rings have the property that every finitely generated module is a direct sum of cyclic modules has been around for many years. We will call these rings FGC rings. The problem originated in I. Kaplanskys papers. For several years, this is one of the major open problems in ring theory.  A deep and difficult study was made by Brandal, Shores-R. Wiegand, S. Wiegand, Wiegand  and V´amos , leading to a complete  solution of the problem in the commutative case. Also, it was shown by Kothe  and Cohen-Kaplansky: “a commutative ring R has the property that every modules is a direct sum of cyclic modules if and only if R is an Artinian principal ideal ring” (recently, a generalization of the Kothe-Cohen-Kaplansky theorem have been given by Behboodi et al. [Proc. Amer. Math. Soc.(2011)(to appear)]. for the non-commutative setting, when all idempotents of R are central). In general, the corresponding problems in the non-commutative case are still open; see [Non-associative algebra and its applications, Lecture Notes in Pure and Applied Mathematics 246 (2006), Appendix B: Unsolved Problems in the Theory of Rings and Modules. Pages 461-516] in which the following problems are considered.

G. Kothe’s Problem: Describe the rings over which each left (or each left, and each right)  module can be decomposed as a direct sum of cyclic modules

I. Kaplansky’s Problem (Reported by A. A. Tuganbaev): Describe the rings over which every finitely generated module can be decomposed as a direct sum of cyclic modules.

The goal of this proposal is to study the structures and  basic properties of FC-pure  injectivity,  FC-pure injective envelope,  I-pure injectivity, FC-pure flatness and I-pure flatness of modules. In particular, we will study the relationship between these notions and  the above two problems of  Kothe and  Kaplansky.  For instance, we prove the following relationship between FC-pure  injectivity and the Kothe and Kaplansky  problems:

Theorem A:   Every left R-module is FC-pure injective (projective) if and only if R is a left Kothe ring (i.e., every left R-module is a direct sum of cyclic modules).

Theorem B:   Every finitely generated left R-module is FC-pure injective (projective) if and only if R is a left FGC- ring (i.e., every finitely generated  left R-module is a direct sum of cyclic modules).

Related Papers:

1- M. Behboodi (Joint with A. Ghorbani, A. Moradzadeh-Dehkordi, and S.H. Shojaee)
On FC-purity and I-purity of modules and Kothe rings
Comm. Algebra 42 (2014), 2061-2081

2- M. Behboodi (Joint with A. Ghorbani, A. Moradzadeh-Dehkordi, and S. H. Shojaee)
C-pure projective modules
Comm. Algebra 41 (2013), 4559-4575

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