On Graded 2-Absorbing Coprimary Submodules
aa r X i v : . [ m a t h . A C ] J a n On Graded -Absorbing Coprimary Submodules Malik
Bataineh and Rashid
Abu-Dawwas
Abstract.
The aim of this article is to introduce the concept of graded 2-absorbing coprimary submodules as a generalization of graded strongly 2-absorbingsecond submodules, and explore some properties of this class. A non-zero graded R -submodule N of a graded R -module M is called a graded 2-absorbing coprimary R -submodule if whenever x, y are homogeneous elements of R and K is a graded R -submodule of M such that xyN ⊆ K , then either x or y is in the graded radicalof ( K : R N ) or xy ∈ Ann R ( N ). Several results have been achieved.
1. Introduction
Throughout this article, G will be a group with identity e and R a commutativering with nonzero unity 1. Then R is said to be G -graded if R = M g ∈ G R g with R g R h ⊆ R gh for all g, h ∈ G where R g is an additive subgroup of R for all g ∈ G .The elements of R g are called homogeneous of degree g . If x ∈ R , then x can bewritten uniquely as X g ∈ G x g , where x g is the component of x in R g . It is known that R e is a subring of R and 1 ∈ R e . The set of all homogeneous elements of R is h ( R ) = [ g ∈ G R g . Let I be an ideal of a graded ring R . Then I is said to be gradedideal if I = M g ∈ G ( I ∩ R g ), i.e., for x ∈ I , x = X g ∈ G x g where x g ∈ I for all g ∈ G . Anideal of a graded ring need not be graded. Assume that M is a left unital R -module.Then M is said to be G -graded if M = M g ∈ G M g with R g M h ⊆ M gh for all g, h ∈ G where M g is an additive subgroup of M for all g ∈ G . The elements of M g are calledhomogeneous of degree g . It is clear that M g is an R e -submodule of M for all g ∈ G .If x ∈ M , then x can be written uniquely as X g ∈ G x g , where x g is the component of x in M g . The set of all homogeneous elements of M is h ( M ) = [ g ∈ G M g . Let N be an R -submodule of a graded R -module M . Then N is said to be graded R -submoduleif N = M g ∈ G ( N ∩ M g ), i.e., for x ∈ N , x = X g ∈ G x g where x g ∈ N for all g ∈ G . Mathematics Subject Classification.
Primary 16W50; Secondary 13A02.
Key words and phrases.
Graded strongly 2-absorbing second submodules; graded 2-absorbingprimary submodules; graded strongly 2-absorbing secondary submodules.
BATAINEH
AND RASHID
ABU-DAWWAS An R -submodule of a graded R -module need not be graded. For more details andterminology, see [
9, 10 ]. Lemma . ( [ ] , Lemma 2.1) Let R be a G -graded ring and M be a G -graded R -module. (1) If I and J are graded ideals of R , then I + J and I T J are graded ideals of R . (2) If N and K are graded R -submodules of M , then N + K and N T K aregraded R -submodules of M . (3) If N is a graded R -submodule of M , r ∈ h ( R ) , x ∈ h ( M ) and I is a gradedideal of R , then Rx , IN and rN are graded R -submodules of M . Moreover, ( N : R M ) = { r ∈ R : rM ⊆ N } is a graded ideal of R . In particular,
Ann R ( M ) = (0 M : R M ) is a graded ideal of R . Let M be a graded R -module, N be a graded R -submodule of M and x ∈ h ( R ). Then ( N : M x ) = { m ∈ M : xm ∈ N } is a graded R -submodule of M , see ([ ], Lemma 2.9). Let P be a proper graded ideal of R . Then the graded radical of P is Grad ( P ), and isdefined to be the set of all r ∈ R such that for each g ∈ G , there exists a positiveinteger n g for which r n g g ∈ P . One can see that if r ∈ h ( R ), then r ∈ Grad ( P ) ifand only if r n ∈ P for some positive integer n . In fact, Grad ( P ) is a graded idealof R , see [ ].In [ ], graded 2-absorbing ideals as a generalization of graded prime ideals havebeen defined. A proper graded ideal P of R is said to be graded 2-absorbing ifwhenever a, b, c ∈ h ( R ) such that abc ∈ P , then either ab ∈ P or ac ∈ P or bc ∈ P .Al-Zoubi and Sharafat in [ ], introduced the concept of graded 2-absorbing primaryideals. This concept generalizes both graded primary ideals and graded 2-absorbingideals. A proper graded ideal P of R is called a graded 2-absorbing primary idealof R if whenever a, b, c ∈ h ( R ) with abc ∈ P , then ab ∈ P or ac ∈ Grad ( P ) or bc ∈ Grad ( P ). Graded second submodules of graded modules over commutativerings were introduced by Ansari-Toroghy and Farshadifar in [ ] as follows: A non-zero graded R -submodule N of a graded R -module M is said to be a graded second R -submodule if for all a ∈ h ( R ), either aN = { } or aN = N . In this case, P = Ann R ( N ) is a graded prime ideal of R , and N is called a graded P -second R -submodule of M . In [ ], a generalization of graded second submodules has beengiven as follows: A non-zero graded R -submodule N of a graded R -module M issaid to be a graded strongly 2-absorbing second R -submodule of M if whenever x, y ∈ h ( R ) and K is a graded R -submodule of M such that xyN ⊆ K , then xN ⊆ K or yN ⊆ K or xy ∈ Ann R ( N ).In this article, we act in accordance with [ ] to introduce the concept of graded2-absorbing coprimary R -submodules of a graded R -module M as a generalizationof graded strongly 2-absorbing second submodules, and explore some properties ofthis class. A non-zero graded R -submodule N of a graded R -module M is calleda graded 2-absorbing coprimary R -submodule if whenever x, y ∈ h ( R ) and K isa graded R -submodule of M such that xyN ⊆ K , then x ∈ Grad (( K : R N )) or y ∈ Grad (( K : R N )) or xy ∈ Ann R ( N ).
2. Graded -Absorbing Coprimary Submodules In this section, we introduce and study the concept of graded 2-absorbing copri-mary submodules.
N GRADED 2-ABSORBING COPRIMARY SUBMODULES 3
Definition . Let M be a graded R -module and N be a non-zero graded R -submodule of M . Then N is said to be a graded -absorbing coprimary R -submoduleof M if whenever x, y ∈ h ( R ) and K is a graded R -submodule of M such that xyN ⊆ K , then x ∈ Grad (( K : R N )) or y ∈ Grad (( K : R N )) or xy ∈ Ann R ( N ) . Clearly, every graded strongly 2-absorbing second submodule is graded 2-absorbingcoprimary. The next example shows that the converse is not true in general.
Example . Consider the Z -module M = Z L Z L Z . Then by ( [ ] , Ex-ample 2.2), N = Z L Z L { } is a -absorbing coprimary submodule of M . So, ifwe consider the trivial graduation for M , N will be a graded -absorbing coprimarysubmodule of M . On the other hand, N is not graded strongly -absorbing secondsubmodule of M by ( [ ] , Corollary 3.4). Proposition . Let M be a graded R -module and N be a graded -absorbingcoprimary R -submodule of M . If K is a graded R -submodule of M such that N * K ,then ( K : R N ) is a graded -absorbing primary ideal of R . Proof.
By Lemma 1.1, ( K : R N ) is a graded ideal of R . Let x, y, z ∈ h ( R )such that xyz ∈ ( K : R N ). Then xyzN = z ( xyN ) ⊆ K , and then xyN ⊆ ( K : M z ).Since N is a graded 2-absorbing coprimary R -submodule of M , x ∈ Grad (( K : M z ) : R N ) or y ∈ Grad (( K : M z ) : R N ) or xy ∈ Ann R ( N ). If xy ∈ Ann R ( N ),then xyN = { } ⊆ K , and then xy ∈ ( K : R N ). If x ∈ Grad (( K : M z ) : R N ),then x n zN ⊆ K for some positive integer n , and then ( xz ) n N ⊆ K , which impliesthat xz ∈ Grad (( K : R N )). Similarly, if y ∈ Grad (( K : M z ) : R N ), then yz ∈ Grad (( K : R N )). Hence, ( K : R N ) is a graded 2-absorbing primary ideal of R . (cid:3) As a direct consequence of Proposition 2.3, we have the following corollary.
Corollary . Let M be a graded R -module and N be a graded -absorbingcoprimary R -submodule of M . Then Ann R ( N ) is a graded -absorbing primary idealof R . Corollary . Let M be a graded R -module and N be a graded -absorbingcoprimary R -submodule of M . Then Grad ( Ann R ( N )) is a graded -absorbing idealof R . Proof.
The result follows by Corollary 2.4 and ([ ], Theorem 2.3). (cid:3) Proposition . Let M be a graded R -module and N be a graded -absorbingcoprimary R -submodule of M . Then aN is a graded -absorbing coprimary R -submodule of M for all a ∈ h ( R ) − Ann R ( N ) . Proof.
Let a ∈ h ( R ) − Ann R ( N ). Then by Lemma 1.1, aN is a graded R -submodule of M . Suppose that x, y ∈ h ( R ) and K is a graded R -submodule of M such that xy ( aN ) ⊆ K . Then xyN ⊆ ( K : M a ). Since N is a graded 2-absorbing coprimary R -submodule of M , x n N ⊆ ( K : M a ) for some positive integer n or y m N ⊆ ( K : M a ) for some positive integer m or xy ∈ Ann R ( N ), and then x n ( aN ) ⊆ K or y m ( aN ) ⊆ K or xy ∈ Ann R ( N ) ⊆ Ann R ( aN ), as needed. (cid:3) Let M and S be two G -graded R -modules. An R -homomorphism f : M → S issaid to be a graded R -homomorphism if f ( M g ) ⊆ S g for all g ∈ G . Lemma . ( [ ] , Lemma 2.16) Suppose that f : M → S is a graded R -homomorphism of graded R -modules. If K is a graded R -submodule of S , then f − ( K ) is a graded R -submodule of M . MALIK
BATAINEH
AND RASHID
ABU-DAWWAS
Lemma . ( [ ] , Lemma 4.8) Suppose that f : M → S is a graded R -homomorphismof graded R -modules. If L is a graded R -submodule of M , then f ( L ) is a graded R -submodule of f ( M ) . Proposition . Suppose that f : M → S is a graded R -homomorphism ofgraded R -modules. (1) If N is a graded -absorbing coprimary R -submodule of M such that N * Ker ( f ) , then f ( N ) is a graded -absorbing coprimary R -submodule of S . (2) If K is a graded -absorbing coprimary R -submodule of S and K ⊆ f ( M ) ,then f − ( K ) is a graded -absorbing coprimary R -submodule of M . Proof. (1) By Lemma 2.8, f ( N ) is a graded R -submodule of S . Let x, y ∈ h ( R ) and K be a graded R -submodule of S such that xyf ( N ) ⊆ K . Then xyN ⊆ f − ( K ). Since f − ( K ) is a graded R -submodule of M by Lemma2.7 and N is a graded 2-absorbing coprimary R -submodule of M , we have x n N ⊆ f − ( K ) for some positive integer n or y m N ⊆ f − ( K ) for somepositive integer m or xy ∈ Ann R ( N ) ⊆ Ann R ( f ( N )), and then x n f ( N ) ⊆ K or y m f ( N ) ⊆ K or xy ∈ Ann R ( f ( N )), as required.(2) By Lemma 2.7, f − ( K ) is a graded R -submodule of M . If f − ( K ) = { } , then since K ⊆ f ( M ), we have f ( f − ( K )) = K = { } , which is acontradiction. So, f − ( K ) = { } . Let x, y ∈ h ( R ) and L be a graded R -submodule of M such that xyf − ( K ) ⊆ L . Since K ⊆ f ( M ), xyK ⊆ f ( L ).Since f ( L ) is a graded R -submodule of S by Lemma 2.8 and K is a graded2-absorbing coprimary R -submodule of S , x n K ⊆ f ( L ) for some positiveinteger n or y m K ⊆ f ( L ) for some positive integer m or xy ∈ Ann R ( K ),and then x n f − ( K ) = f − ( x n K ) ⊆ L or y m f − ( K ) = f − ( y m K ) ⊆ L or xy ∈ Ann R ( K ) = Ann R ( f − ( K )), as needed. (cid:3) Lemma . Let M be a graded R -module and N be a non-zero graded R -submodule of M . Then N is a graded -absorbing coprimary R -submodule of M ifand only if for any x, y ∈ h ( R ) , either x n N ⊆ xyN for some positive integer n or y m N ⊆ xyN for some positive integer m or xy ∈ Ann R ( N ) . Proof.
Suppose that N is a graded 2-absorbing coprimary R -submodule of M .Let x, y ∈ h ( R ). Then xy ∈ h ( R ), and then by Lemma 1.1, xyN is a graded R -submodule of M such that xyN ⊆ xyN . Since N is a graded 2-absorbing coprimary R -submodule of M , x n N ⊆ xyN for some positive integer n or y m N ⊆ xyN forsome positive integer m or xy ∈ Ann R ( N ). The converse is clear. (cid:3) Let M be a graded R -module and S be a multiplicative subset of h ( R ). Then S − M is a graded S − R -module with( S − M ) g = n ms , m ∈ M h , s ∈ S ∩ R hg − o , ( S − R ) g = n as , a ∈ R h , s ∈ S ∩ R hg − o . Proposition . Let M be a graded R -module, S be a multiplicative subsetof h ( R ) and N be a graded R -submodule of M such that S − N = { } . If N is agraded -absorbing coprimary R -submodule of M , then S − N is a graded -absorbingcoprimary S − R -submodule of S − M . N GRADED 2-ABSORBING COPRIMARY SUBMODULES 5
Proof.
Let x, y ∈ h ( R ) and s, t ∈ S . Since N is a graded 2-absorbing copri-mary R -submodule of M , by Lemma 2.10, x n N ⊆ xyN for some positive integer n or y m N ⊆ xyN for some positive integer m or xy ∈ Ann R ( N ). If xy ∈ Ann R ( N ),then xs yt ( S − N ) = { } . If x n N ⊆ xyN , then ( xs ) n ( S − N ) = ( x n s n )( S − N ) = s n S − ( x n N ) ⊆ s n S − ( xyN ) = S − ( xyN ) = ( s t )( S − ( xyN )) = ( xs yt )( S − N ). Sim-ilarly, if y m N ⊆ xyN , then ( xs ) m ( S − N ) ⊆ ( xs yt )( S − N ). Hence, by Lemma 2.10, S − N is a graded 2-absorbing coprimary S − R -submodule of M . (cid:3) Definition . Let M be a G -graded R -module, g ∈ G and N be a non-zerograded R -submodule of M . Then N is said to be a g - -absorbing coprimary R -submodule of M if whenever x, y ∈ R g and K is a graded R -submodule of M such that xyN ⊆ K , then x ∈ Grad (( K : R N )) or y ∈ Grad (( K : R N )) or xy ∈ Ann R ( N ) . Lemma . Let M be a G -graded R -module, g ∈ G , I be a graded ideal of R and N be a g - -absorbing coprimary R -submodule of M . If x ∈ R g and K isa graded R -submodule of M such that IxN ⊆ K , then x ∈ Grad (( K : R N )) or I g ⊆ Grad (( K : R N )) or I g x ⊆ Ann R ( N ) . Proof.
Suppose that x / ∈ Grad (( K : R N )) and I g x * Ann R ( N ). Then thereexists y ∈ I g such that xy / ∈ Ann R ( N ), and then xyN ⊆ K . Since N is g -2-absorbing coprimary, y ∈ Grad (( K : R N )). Let z ∈ I g . Then ( y + z ) xN ⊆ K , andthen y + z ∈ Grad (( K : R N )) or ( y + z ) x ∈ Ann R ( N ). If y + z ∈ Grad (( K : R N )),then z ∈ Grad (( K : R N )). If ( y + z ) x ∈ Ann R ( N ), then xz / ∈ Ann R ( N ). So, zxN ⊆ K implies that z ∈ Grad (( K : R N )). Hence, I g ⊆ Grad (( K : R N )). (cid:3) Theorem . Let M be a G -graded R -module, g ∈ G , I, J be two gradedideals of R and N be a g - -absorbing coprimary R -submodule of M . If K is agraded R -submodule of M such that IJ N ⊆ K , then I g ⊆ Grad (( K : R N )) or J g ⊆ Grad (( K : R N )) or I g J g ⊆ Ann R ( N ) . Proof.
Suppose that I g * Grad (( K : R N )) and J g * Grad (( K : R N )). Thenthere exist x ∈ I g − Grad (( K : R N )) and y ∈ J g − Grad (( K : R N )). Let z ∈ I g and w ∈ J g . Now, xJ N ⊆ K , so by Lemma 2.13, xJ g ⊆ Ann R ( N ), and so( I g − Grad (( K : R N ))) J g ⊆ Ann R ( N ). Similarly, I g y ⊆ Ann R ( N ), and so I g ( J g − Grad (( K : R N ))) ⊆ Ann R ( N ). So, we conclude that xy, xw, zy ∈ Ann R ( N ). Now,( x + z )( y + w ) N ⊆ K . Therefore, x + z ∈ Grad (( K : R N )) or y + w ∈ Grad (( K : R N ))or ( x + z )( y + w ) ∈ Ann R ( N ). If x + z ∈ Grad (( K : R N )), then z / ∈ Grad (( K : R N )).Thus z ∈ I g − Grad (( K : R N )), which implies that zw ∈ Ann R ( N ). Similarly, if y + w ∈ Grad (( K : R N )), then zw ∈ Ann R ( N ). If ( x + z )( y + w ) = xy + xw + yz + zw ∈ Ann R ( N ), then zw ∈ Ann R ( N ). Hence, I g J g ⊆ Ann R ( N ). (cid:3) Graded comultiplication modules have been introduced by Toroghy and Far-shadifar in [ ]; a graded R -module M is said to be graded comultiplication if forevery graded R -submodule N of M , N = (0 : M I ) for some graded ideal I of R , orequivalently, N = (0 : M Ann R ( N )). Proposition . Let M be a graded comultiplication R -module. If N isa graded -absorbing coprimary R -submodule of M such that Grad ( Ann R ( N )) = Ann R ( N ) , then N is a graded strongly -absorbing second R -submodule of M . Inparticular, N is a graded -absorbing second R -submodule of M . Proof.
The result follows by Corollary 2.5 and ([ ], Proposition 3.7). (cid:3) MALIK
BATAINEH
AND RASHID
ABU-DAWWAS
Let R and R be two G -graded rings, M be a G -graded R -module, M be a G -graded R -module, R = R × R and M = M × M . Then M is a G -graded R -module by M g = ( M ) g × ( M ) g for all g ∈ G . Theorem . Let R and R be two G -graded rings, M be a G -graded R -module, M be a G -graded R -module, R = R × R and M = M × M . (1) If N = N × N is a graded -absorbing coprimary R -submodule of M , then Ann R ( N ) is a graded primary ideal of R and Ann R ( N ) is a gradedprimary ideal of R . (2) If N is a graded R -submodule of M , N is a graded R -submodule of M such that Ann R ( N ) is a graded primary ideal of R , Ann R ( N ) is agraded primary ideal of R and N = N × N , then Ann R ( N ) is a graded -absorbing primary ideal of R . (3) If N is a graded -absorbing coprimary R -submodule of M and N = N × { M } , then Ann R ( N ) is a graded -absorbing primary ideal of R . (4) If N is a graded -absorbing coprimary R -submodule of M and N = { M } × N , then Ann R ( N ) is a graded -absorbing primary ideal of R . Proof. (1) By ([ ], Lemma 3.12), N is a graded R -submodule of M and N is a graded R -submodule of M , and then Ann R ( N ) is a graded ideal of R and Ann R ( N ) is a graded ideal of R . Since N is graded 2-absorbingcoprimary, by Corollary 2.4, Ann R ( N ) = Ann R ( N ) × Ann R ( N ) is agraded 2-absorbing primary ideal of R , and then by ([ ], Theorem 2.10), Ann R ( N ) is a graded primary ideal of R and Ann R ( N ) is a gradedprimary ideal of R .(2) By ([ ], Lemma 3.12), N is a graded R -submodule of M , and then Ann R ( N )is a graded ideal of R . Since Ann R ( N ) is a graded primary ideal of R , Ann R ( N ) × R is a graded primary ideal of R . Similarly, R × Ann R ( N ) is a graded primary ideal of R . So, by ([ ], Lemma 2.9), Ann R ( N ) T Ann R ( N ) = Ann R ( N ) × Ann R ( N ) = Ann R ( N ) is agraded 2-absorbing primary ideal of R .(3) By ([ ], Lemma 3.12), N is a graded R -submodule of M , and then Ann R ( N )is a graded ideal of R . Since N is graded 2-absorbing coprimary, by Corol-lary 2.4, Ann R ( N ) is a graded 2-absorbing primary ideal of R , and then Ann R ( N ) × R = Ann R ( N ) is a graded 2-absorbing primary ideal of R .(4) This result holds similar to part (3). (cid:3) References [1] R. Abu-Dawwas, M. Bataineh and H. Shashan, Graded generalized 2-absorbing submod-ules, Beitr¨age zur Algebra und Geometrie / Contributions to Algebra and Geometry, (2020),https://doi.org/10.1007/s13366-020-00544-1.[2] K. Al-Zoubi, R. Abu-Dawwas and S. C¸ eken, On graded 2-absorbing and graded weakly 2-absorbing ideals, Hacettepe Journal of Mathematics and Statistics, 48 (3) (2019), 724-731.[3] K. Al-Zoubi and N. Sharafat, On graded 2-absorbing primary and graded weakly 2-absorbingprimary ideals, Journal of the Korean Mathematical Society, 54 (2) (2017), 675-684.[4] H. Ansari-Toroghy and F. Farshadifar, Graded comultiplication modules, Chiang Mai Journalof Science, 38 (3) (2011), 311-320.[5] H. Ansari-Toroghy and F. Farshadifar, On graded second modules, Tamkang Journal of Math-ematics, 43 (4) (2012), 499-505.[6] S. E. Atani and F. E. K. Saraei, Graded modules which satisfy the gr-radical formula, ThaiJournal of Mathematics, 8 (1) (2010), 161-170.
N GRADED 2-ABSORBING COPRIMARY SUBMODULES 7 [7] S. C¸ eken, 2-Absorbing coprimary submodules, Beitr¨age zur Algebra und Geometrie / Contri-butions to Algebra and Geometry, (2021), https://doi.org/10.1007/s13366-020-00554-z.[8] F. Farzalipour, P. Ghiasvand, On the union of graded prime submodules, Thai Journal ofMathematics, 9 (1) (2011), 49-55.[9] R. Hazrat, Graded rings and graded Grothendieck groups, Cambridge University press, 2016.[10] C. Nastasescu and F. Oystaeyen, Methods of graded rings, Lecture Notes in Mathematics,1836, Springer-Verlag, Berlin, 2004.[11] M. Refai and R. Abu-Dawwas, On generalizations of graded second submodules, ProyeccionesJournal of Mathematics, 39 (6) (2020), 1547-1564.[12] M. Refai, M. Hailat and S. Obiedat, Graded radicals and graded prime spectra, Far EastJournal of Mathematical Sciences, (2000), 59-73.[13] H. Saber, T. Alraqad and R. Abu-Dawwas, On graded s -prime submodules, AIMS Mathe-matics, 6 (3) (2020), 2510-2524. Department of Mathematics and Statistics, Jordan University of Science andTechnology, Irbid, Jordan
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