Studying Some Results about Completely Coretractable Ring (CC-ring)

Abstract

In this paper , some results about the concept of completely coretractable ring are studied and also another results about the coretractable module are recalled and investigated.

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Global Journal of Mathematics Vol. 10, No.1, May 29, 2017
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Studying Some Results about Completely Coretractable Ring (CC-ring)
Inaam Mohammed Ali Hadi 1 , Shukur Neamah Al-aeashi 2
1st Department Of Mathematics, College 0f Education for Pure Sciences (Ibn-Al-Haitham) , University Of
Baghdad , Iraq
2nd Department Of Urban Planning , College 0f Physical Planning, University Of Kufa , Iraq
Abstract
In this paper , some results about the concept of completely coretractable ring are studied and also another results about
the coretractable module are recalled and investigated.
Indexing terms/Keywords : coretractable module , completely coretractable ring.
SUBJECT CLASSIFICATION 16D10; 16D40.
INTRODUCTION
Throughout this paper , R is a ring with unity and all modules are unitary right R-modules . Amini in [1] introduced the
concept of coretractable modules , where an R-module M is called coretractable if for each a proper submodule N of M ,
there exists a nonzero R-homomorphism f:M/N→M (that is f HomR(M/N,M) ) . After that many authors investigate some
results and give another generalize about of this class see [2],[3],[4],[5] . Also Amini in [1] introduced the concept of
completely coretractable ring , where a ring R is called completely coretractable ring if for each R-module is
coretractable after that some authors study this concept see Zemlicka in [6]. It is clear that the class of strongly
coretractable module contains the class of coretractable , but not conversely for example the Z-module Z4 is coretractable
but not strongly coretractable module as explains in [3] farther the class of cortractable implies automatically into the
classes P-coretractable and Y-coretractable modules see [4],[5] . Many characterizations about these generalizations are
studied there . Our aim in this paper is to study the completely cortractable ring ( up to knowledge ) where we introduce
some results about this class concerned with anther related concepts . In Section one of this work is devoted to recall
some basic properties of coretractable modules . Also we added some new results ( we could prove it ) . Also many
connections between it and other classes of modules were given .
1. ON CORETRACTABLE MODULES
First , we recall by the following definition :
Definition(1.1) [1] : An R-module M is called coretractable if for each a proper submodule N of M , there exists a nonzero
R-homomorphism f:M/N→M.
Examples and Remarks (1.2):
(1) An R-module M is coretractable if and only if for each proper submodule N of M , there exists a nonzero mapping
f EndR(M) such that f(N)=0 ; that is N kerf.
(2) Clearly every semisimple module is coretractable , and hence every R-module over a semisimple ring is
coretractable [1] .
But it may be that coretractable module not semisimple as the Z-module Z4 .
(3) Coretractability is preserved by an isomorphism .
Proposition(1.3): An R-module M is coretractable if and only if HomR(M/N, M) 0 for any proper essential submodule N of
M .
Proposition(1.4): An R-module M is coretractable if and only if M is coretractable -module ( where =R/annM ) .
Recall that an R-module M is called cogenerator if for every nonzero homomorphism f:M1→ M2 where M1 and M2 are R-
modules , g:M2→M such that g◦f ≠ 0 [11, P.507] and [8, P.53] . Equivalently an R-module M is called a
cogenerator if for any R-module N and 0 x N, there exists g: N M such that g(x) ≠0 [11 , P.507] .
Proposition(1.5): [1] Every cogenerator R-module is coretractable module .
Recall that ( Schur’s Lemma ) stated that " If M is simple module , then S=End(M) is a division ring " [12,P.168] .
Proposition(1.6): An R-module M is simple if and only if M is a coretractable and EndR(M) is a division ring .
Proof : ( ) Since M is simple , it is clear that M is a coretractable module, EndR(M) is division ring by Schur’s Lemma .
( ) Let 0≠K<M . As M is coretractable module , then there exists f : M M , f≠0 such that f(K)=0, hence f is not one-one
which is a contradiction with the hypothesis EndR(M) is a division. Thus M is simple module .
Recall that " An R-module M is called quasi-Dedekind if every proper nonzero submodule N of M is quasi-invertible
where a submodule N of M is called quasi-invertible if HomR(M/N,M)=0 " [13] . A nonzero ideal ( right ideal) I of a ring R

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is quasi-invertible ideal (right ideal) of R if I is quasi-invertible submodule of R . Also " M is a quasi-Dedekind R-module if
for any nonzero f EndR(M) , f is monomorphism ; that is kerf= (0) " [13,Theorem(1.5) , P.26] .
Hence we get the following remarks for an R-module M .
Remarks(1.7):
(1) An R-module M is a coretractable if and only if each proper submodule of M is not quasi-invertible submodule .
(2) For a ring R , R is a coretractable R-module if and only if annJ 0 for each nonzero proper right ideal J of R .
(3) Every integral domain (not simple) is not coretractable ring .
(4) An R-module M is coretractable quasi-Dedekind if and only if M is simple module .
(5) Let M be not simple R-module . If M is a coretractable module , then M is not quasi-Dedekind .
(6) Let M be an R-module . If M is a nonsingular uniform module , then M is a quasi-Dedekind , and hence M is not
coretractable .
Proof : Let N be a proper nonzero submodule of M , so N is essential submodule of M ( since M is uniform ) , hence M/N
is singular . But M is a nonsingular . Thus Hom(M/N,M)=0 ; that is N is quasi-invertible submodule . Therefore M is quasi-
Dedekind . Therefore M is not coretractable module .
Recall that " A submodule N of M is called coquasi-invertible submodule of M if HomR(M,N) = 0 " [14,P.8] and " A
nonzero R-module M is called coquasi-Dedekind module if every proper submodule of M is coquasi-invertible module of
M " [14,P.32] . Equivalently , " M is coquasi-Dedekind module if for each f EndR(M), f≠0 , f is an epimomorphism " . [14 ,
Theorem(2.1.4) ,P.33] .
Proposition(1.8): Let M be a coretractable R-module . Then the following statements are equivalent :
(1) M is a simple module ;
(2) M is a quasi-Dedekind module ;
(3) M is a coqusi- Dedekind module and Rad(M)=(0) ;
(4) EndR(M) is a division ring .
Proof :
(1) (4) It follows by Proposition(1.6) .
(1) (2) It follows by Remark(1.7(4)) .
(1) (3) It is clear .
(3) (1) Since Rad(M)=0 , M has a maximal submodule say N , so N is a proper submodule of M . As M is coretractable
module , then there exists f :M M , f≠0 and f(N)=0 , then N kerf , but N is maximal so that N=kerf . On the other
hand, by 1st fundamental theorem M/kerf = M/N f(M) , but f(M)= M since M is coquasi-Dedekind module , then M/N M .
Therefore M is simple module since M/N is simple .
Recall that " An R-module M is called retractable if Hom(M,N) 0 for each nonzero submodule N of M " [15] .
Examples and Remarks (1.9):
(1) Zn as Z-module is retractable for each positive integer n>1 .
(2) Every semisimple module is retractable module .
(3) " Every commutative ring with identity is a retractable " , hence a retractable R-module M may be not
coretractable . As the Z-module Z is retractable and it is not coretractable .
(4) A coretractable module may be not retractable . For example Zp∞ as Z-module is coretractable, but it is not
retractable see [16,P.71] .
Corollary(1.10): Let M be a retractable and coretractable module , then the following statements are equivalent :
(1) S has no zero divisors ;
(2) M is a simple module ;
(3) S is a division ring .
Proof :
(1) (2) Suppose that there exists a proper K of M and K≠0 . Hence Hom(M,K)=0 by Proposition(1.8) which is a
contradiction with M is retractable module . Then K=0 and hence M is simple module.
(2) (3) Since M is simple , then S is a division ring by Schur's Lemma[12,P.168]
(3) (1) It is clear .

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2. COMPLETELY CORETRACTABLE RING ( CC-RING)
For a ring R , recall that " If every R-module is coretractable , then R is called completely coretractable ring ( briefly
CC-ring )" [1] . We review some important properties . Moreover we add many results related with this concept .
Examples and Remarks (2.1):
(1) " Every semisimple ring is a completely coretractable (CC-ring) " [1] .
(2) If R is a Kasch ring and J(R)=0 , then R is a CC-ring .
Proof :
By [1,Lemma(3.2)] , R is a semisimple and hence R is a CC-ring by part (1)
(3) If R is a Kasch regular ring , then R is a CC-ring .
Proof :
Since R is regular , so J(R)=0 and hence R is a CC-ring by part (2) .
Recall that " A ring R is semi-Artinian if every R-module has a minimal submodule ( equivalently, if every cyclic R-
module has a minimal submodule ; see [10, Proposition VIII.2.5] ) " .
"Proposition(2.2): Let R be a semi-Artinian max ring . If R has a unique simple R-module , then R is a CC-ring "
[1,Proposition(3.5)] .
"Proposition(2.3): Let R be a max ring . If every cyclic R-module is coretractable , then R is a CC-ring " [1,
Proposition(3.6)] .
"Proposition(2.4): Let R be a CC-ring . Then an R-module M is a Noetherian if and only if it is an Artinian "
[1,Corollary(3.11)] .
Recall that " Let M be a right R-module and let S = EndR(M). Then M is called a dual-Rickart (or a d-Rickart) module if
the image in M of any single element of S is generated by an idempotent of S. Equivalently , M is called d-Rickart module
if for all f S , Imf M "[7] .
"Proposition(2.5): A ring R is a dual Rickart ring if and only if R is semisimple ring " [9] .
Corollary(2.6): If R is a dual Rickart ring , then R is a CC-ring .
Proof: Since R is dual Rickart , then R is semisimple and hence R is CC-ring by Examples and Remarks (2.1(1)) .
Now , we present
Proposition(2.7): Let R be a ring with J(R)=0 , then the following statements are equivalent :
(1) R is a coretractable ring ;
(2) R is a semisimple ring ;
(3) R is a CC-ring ;
(4) All free R-module is a coretractable ;
(5) All finitely generated free R-module is a coretractable .
Proof : (1) (2) It follows by [1,Lemma(3.7)] .
(2) (3) By Examples and Remarks (2.1(1)) .
(3) (4) (5) It is clear .
(5) (1) It is clear since R is finitely generated free R-module , so R is coretractable
Corollary(2.8): Let R be a commutative regular ring in sense of Von Neumann , then the following statements are
equivalent :
(1) R is a coretractable ring ;
(2) R is a semisimple ring ;
(3) R is a CC-ring ;
(4) All free R-module are coretractable ;
(5) All finitely generated free R-module are coretractable .
Proof: Since R is a commutative regular ring , then J(R)=0 . Thus the result follows by Proposition(2.7) .
Corollary(2.9): Let R be a commutative Von Neumann regular ring . If R is coretractable , then R is a principal ideal ring .
Proof :

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Since R is a commutative regular coretractable ring , then by Corollary(2.8) , then R is semisimple . Hence every ideal
generated by idempotent . Thus R is a principal ideal ring .
Proposition(2.10): Let R be a max ring . Then R is a coretractable ring if and only if every free R-module is a
coretractable .
Proof : ( ) Since R is a coretractable ring and R is a max ring , then by [1, Proposition(2.7)] , Ri is also coretractable
(where Ri=R and I is an index set) . Let F be any free R-module . Then F Ri (where Ri=R and I is an index set) .
Thus F is coretractable
( ) It is clear .
Corollary(2.11): Let R be a coretractable max ring with J(R)=0 , then R is a CC- ring
Proof :
Since R is a max ring , so by Proposition(2.10) , every free R-module is coretractable. Then by Proposition(2.7), hence R
is a CC-ring .
Proposition(2.12): Let R be a commutative ring such that R/annM is a coretractable ring . Then every cyclic R-module is
a coretractable .
Proof : Let M be a cyclic R-module . Then M is a faithful -module where =R/annM , so that M . But is
coretractable -module , hence by Proposition (1.4) . M is a coretractable -module and so M is a coretractable R-module
by Proposition (1.4) .
Corollary(2.13 ): Let R be a commutative ring . Then the following statements are equivalent :
(1) R is a CC-ring ;
(2) Every cyclic R-module is a coretractable ;
(3) R/annM is a coretractable ring .
Proof :
(1) (2) It follows by [1,Theorem(3.14)] .
(3) (2) It follows by Proposition(2.12 ) .
(2) (3) Since R/annM is a cyclic R-module , so by part(2) R/annM coretractable R-module , hence R/annM is a
coretractable -module (where =R/annM) ; that is R/annM is a coretractable ring .
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