Article
Kyungpook Mathematical Journal 2021; 61(2): 239-248
Published online June 30, 2021
Copyright © Kyungpook Mathematical Journal.
Weak FI-extending Modules with ACC or DCC on Essential Submodules
Adnan Tercan, Ramazan Yaşar*
Department of Mathematics, Hacettepe University, Beytepe Campus, Ankara 06532, Turkey
e-mail : tercan@hacettepe.edu.tr
Hacettepe-ASO 1.OSB Vocational School, Hacettepe University, 06938 Sincan Ankara, Turkey
e-mail : ryasar@hacettepe.edu.tr
Received: April 19, 2020; Accepted: December 14, 2020
Abstract
In this paper we study modules with the WFI+-extending property. We prove that if M satisfies the WFI+-extending, pseudo duo properties and M/(Soc M) has finite uniform dimension then M decompose into a direct sum of a semisimple submodule and a submodule of finite uniform dimension. In particular, if M satisfies the WFI+-extending, pseudo duo properties and ascending chain (respectively, descending chain) condition on essential submodules then M=M1 ⊕ M2 for some semisimple submodule M1 and Noetherian (respectively, Artinian) submodule M2. Moreover, we show that if M is a WFI-extending module with pseudo duo, C2 and essential socle then the quotient ring of its endomorphism ring with Jacobson radical is a (von Neumann) regular ring. We provide several examples which illustrate our results.
Keywords: CS-module, uniform dimension, ascending chain condition on essential submodules, FI-extending, WFI-extending
1. Introduction
Assume that all rings are associative and have identity elements and all modules are unital right modules. Let
Armendariz [2, Proposition 1.1] proved that a module
-
(i)
M/N has finite uniform dimension for every essential submoduleN ofM ,
-
(ii) every homomorphic image of
has finite uniform dimension.
Camillo and Yousif [6, Corollary 3] proved that if
A module
A module
A module
No other implications can be added to this table in general. To see why this is the case, please consult [19]. Note that it is an open problem to determine whether the
The purpose of this paper is to try to extend the result of [16, Theorem 11, Corollary 12] to
2. Weak FI+ -extending Modules
Let
Example 2.1.
Let
For more examples similar to Example 2.1 (see [17, Corollary 16]). Surprisingly, Example 2.1 and [17, Corollary 16] also show that we can not replace
Theorem 2.2.
Let
Now
The next example shows that
Example 2.3.
Let Kbe a field and Van infinite dimensional vector space over K. Let Rbe the trivial extension Kwith Vi.e. ,
Then
Corollary 2.4.
Let
Now, let us think of general modules over arbitrary rings. Since we require both the pseudo duo property and that
Example 2.5.
-
(i) Let
M be the free-module of infinite rank i.e., . Then . Hence M satisfies pseudo duo property. However,which has infinite uniform dimension.
-
(ii) Let
R be a prime ring and let. Then, it is clear that which is essential in MR and hencehas finite uniform dimension. Now, let . Define by and by . Obviously f1 ,. Let , . So, we have and . It follows that MR does not have pseudo duo property.
The following is a key lemma for our main theorem in this section.
Lemma 2.6.
Let
Now we have the following result which was pointed out in the introduction.
Theorem 2.7.
Let
Now assume that
Next we apply the former result to
Corollary 2.8.
Let
Recall that a module
Corollary 2.9.
Let
Corollary 2.10.
Let
We close this section by giving an example which illustrates that the converse of Theorem 2.7 is not true, in general.
Example 2.11.
Let
Now, let
3. Endomorphism Rings of Weak FI -extending Modules
In this section our concern is the endomorphism ring of weak
Theorem 3.1.
Let
Next assume that
In any case, we have that
Corollary 3.2.
Let
Corollary 3.3.
Let
Since
Note that there are commutative, local rings
In the sense of construction certain examples, Corollary 3.3 is a useful tool. For example, let
Furthermore, the next example shows that the pseudo duo assumption in Theorem 3.1 is not unnecessary either.
Example 3.4.
Let
References
- F. W. Anderson and K. R. Fuller, Rings and categories of modules, Graduate Texts in Mathematics 13, Springer Science & Business Media, 2012.
- E. P. Armendariz, Rings with DCC on essential left ideals, Comm. Algebra, 8(1980), 299-308.
- G. F. Birkenmeier, G. CĂlugĂreanu, L. Fuchs and H. P. Goeters, The fully invariant extending property for abelian groups, Comm. Algebra, 29(2001), 673-685.
- G. F. Birkenmeier, B. J. Müller and S. T. Rizvi, Modules in which every fully invari-ant submodule is essential in a direct summand, Comm. Algebra, 30(2002), 1395-1415.
- G. F. Birkenmeier, J. K. Park and S. T. Rizvi, Extensions of rings and modules, Birkhuser, New York, 2013.
- V. Camillo and M. F. Yousif, CS-modules with ACC or DCC on essential submodules, Comm. Algebra, 19(1991), 655-662.
- N. V. Dung, D. V. Huynh, P. F. Smith and R. Wisbauer, Extending modules, Pitman, London, 1994.
- K. R. Goodearl, Singular torsion and the splitting properties, Memoirs of the Amer-ican Mathematical Society 124, Amer. Math. Soc., 1972.
- I. Kaplansky, Infinite abelian groups, University of Michigan Press, Ann Arbor, 1969.
- S. H. Mohamed and B. J. Müller, Continuous and discrete modules, London Math-ematical Society Lecture Note Series 147, Cambridge University Press, Cambridge, 1990.
- W. K. Nicholson and M. F. Yousif, Quasi-frobenius rings, Cambridge Tracts in Mathematics 158, Cambridge University Press, 2003.
- P. F. Smith, CS-modules and weak CS-modules, Non-commutative ring theory, 99-115, Lecture Notes in Math. 1448, Springer, Berlin, Heidelberg, 1990.
- P. F. Smith, Modules with many direct summands, Osaka J. Math., 27(1990), 253-264.
- P. F. Smith and A. Tercan, Generalizations of CS-modules, Comm. Algebra, 21(1993), 1809-1847.
- A. Tercan, On the endomorphism ring of modules with (C11) and (C2), Hacettepe Bull. Nat. Sci. Engrg., 22(1993), 1-7.
- A. Tercan, Weak (C11+)-modules with ACC or DCC on essential submodules, Tai-wanese J. Math., 5(2001), 731-738.
- A. Tercan, Weak C11 modules and algebraic topology type examples, Rocky Mountain J. Math., 34(2004), 783-792.
- A. Tercan and C. C. Yücel, Module theory, extending modules and generalizations, Birkhäuser-Springer, Basel, 2016.
- R. Ya¸sar, Modules in which semisimple fully invariant submodules are essential in summands, Turkish J. Math., 43(2019), 2327-2336.
- J. M. Zelmanowitz, Endomorphism rings of torsionless modules, J. Algebra, 5(1967), 325-341.