Kyungpook Mathematical Journal 2020; 60(1): 44-51
Published online March 31, 2020
Copyright © Kyungpook Mathematical Journal.
On Generalized FI-extending Modules
Canan Celep Yücel
Department of Mathematics, Faculty of Arts and Sciences, Pamukkale University, Denizli 20070, Turkey
e-mail : email@example.com
Received: June 3, 2016; Revised: November 2, 2019; Accepted: December 3, 2019
Keywords: fully invariant submodule, FI-extending, GFI-extending.
In recent years, the theory of extending modules and rings has come to play an important role in the theory of rings and modules. A module
In , the following statements were proved: (1) Any direct sum of FI-extending modules is FI-extending; (2) A ring
In this work, we determine a generalization of the FI-extending module which is not only preserved under various extensions including direct sums and matrix constructions. We define a module
Throughout this paper, all rings are associative with unity and all modules are unital right modules. Recall from , a submodule
2. GFI-extending Modules
In this section, we begin with the definition of the main concept of this paper. Then, we study relationships between the extending, FI-extending and GFI-extending modules. We also consider connections between the singular and GFI-extending modules.
M is extending, M is FI-extending, M is GFI-extending.
(i) ⇒ (ii) and (ii) ⇒ (iii). These implications are clear.
This proof is routine.
In general, the reverse implication of the above result does not hold. For example, let ℤ be the ring of all integers. Then, ℤ is extending. Hence, it is FI-extending. By Proposition 2.2, ℤ is GFI-extending as a right ℤ-module, but it is nonsingular.
The following lemma gives the equivalence of the FI-extending and GFI-extending modules.
The following proposition is a consequence of Definition 2.1 and gives a characterization of the GFI-extending modules by fully invariant submodules.
M is GFI-extending. For any fully invariant submodule N of M, M has a decomposition M= K⊕ K′ such that N≤ K and that M/( K′+ N) is singular. For any fully invariant submodule N of M, M/N has a decomposition M/N= K/N⊕ K′/N such that K is a direct summand of M and that M/K′ is singular. For any fully invariant submodule N of M, there is a direct summand K of M such that for any x∈ K there is an essential right ideal I of R such that xI≤ N.
The proof is straightforward.
The next result establishes connections between a GFI-extending module and its injective hull.
(⇒). Assume that
In the next lemma and theorem, we prove that the GFI-extending property of a ring
The proof is a clear consequence of [2, Lemma 2.2] because
3. Direct Sum of GFI-Extending Modules
In , it was proved that a direct sum of FI-extending modules is FI-extending. It is also known that a direct sum of singular modules is singular . In this section, we show that GFI-extending property is closed under direct sums.
Suppose the modules
M is finitely generated. M is of bounded order (i.e., nM= 0, for some positive integer n). M is divisible.
(i) Every finitely generated Abelian group is a direct sum of uniform ℤ-modules.
(ii) This part is from [10, p.262].
The following corollaries are direct consequences of Theorem 3.1.
The proof is clear by Theorem 3.1.
In general, it is not known whether a direct summand of a GFI-extending module is also GFI-extending. However, in the following theorem, we prove that this implication is true under some restrictions.
By Lemma 2.6,
The author gratefully thanks to the Referee for the constructive comments and recommendations which definitely help to improve the readability and quality of the paper.
- FW. Anderson, and KR. Fuller. Rings and categories of modules,
, Springer Verlag, New York-Heidelberg, 1974.
- GF. Birkenmeier, BJ. Müller, and ST. Rizvi.
Modules in which every Fully Invariant Submodule is essential in a direct summand. Comm Algebra., 30(3)(2002), 1395-1415.
- GF. Birkenmeier, and A. Tercan.
When some complement of a submodule is a summand. Comm Algebra., 35(2007), 597-611.
- GF. Birkenmeier, A. Tercan, and CC. Yücel.
The extending condition relative to sets of submodules. Comm Algebra., 42(2014), 764-778.
- NV. Dung, DV. Huynh, PF. Smith, and R. Wisbauer. Extending modules,
, Longman Scientific and Technical, England, 1994.
- C. Faith. Algebra II: ring theory,
, Springer Verlag, Berlin-New York, 1976.
- KR. Goodearl. Ring theory: non-singular rings and modules,
, Marcel Dekker, Inc, New York, 1976.
- KR. Goodearl. Von Neumann regular rings,
, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979.
- Z. Quing-yi.
On generalized extending modules. J Zhejiang Univ Sci A., 8(6)(2007), 939-945.
- JJ. Rotman. An introduction to the theory of groups. 3rd ed,
, Allyn and Bacon Inc., Boston, MA, 1984.
- PF. Smith, and A. Tercan.
Generalizations of CS-modules. Comm Algebra., 21(1993), 1809-1847.
- A. Tercan, and CC. Yücel. Module theory, extending modules and generalizations,
, Birkhäuser/Springer, Switzerland, 2016.