On Graded 1-Absorbing Prime Submodules
aa r X i v : . [ m a t h . A C ] J a n On Graded -Absorbing Prime Submodules Ahmad
Ka’abneh and Rashid
Abu-Dawwas
Abstract.
Let G be a group with identity e , R be a commutative G -gradedring with unity 1 and M be a G -graded unital R -module. In this article, weintroduce the concept of graded 1-absorbing prime submodule. A proper graded R -submodule N of M is said to be a graded 1-absorbing prime R -submodule of M if for all non-unit homogeneous elements x, y of R and homogeneous element m of M with xym ∈ N , either xy ∈ ( N : R M ) or m ∈ N . We show that the newconcept is a generalization of graded prime submodules at the same time it is aspecial graded 2-absorbing submodule. Several properties of a graded 1-absorbingprime submodule have been obtained. We investigate graded 1-absorbing primesubmodules when the components { M g : g ∈ G } are multiplication R e -modules.
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 . Moreover, R e isa subring of R and 1 ∈ R e and h ( R ) = [ g ∈ G R g . Let I be an ideal of a graded ring R . Then I is said to be graded ideal 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 . An ideal 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 subgroupof M for all g ∈ G . The elements of M g are called homogeneous of degree g . Itis clear that M g is an R e -submodule of M for all g ∈ G . If x ∈ M , then x canbe written uniquely as X g ∈ G x g , where x g is the component of x in M g . Moreover, h ( M ) = [ g ∈ G M g . Let N be an R -submodule of a graded R -module M . Then N is Mathematics Subject Classification.
Primary 16W50; Secondary 13A02.
Key words and phrases.
Graded 1-absorbing prime ideal; graded 1-absorbing prime submodule;graded prime submodule.
KA’ABNEH
AND RASHID
ABU-DAWWAS said to be graded R -submodule if 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 . An R -submodule of a graded R -module need not begraded.For more details and terminology, see [ ] and [ ]. 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 . Similarly, if M is a graded R -module, N a graded R -submodule of M and m ∈ h ( M ), then ( N : R m ) is a graded ideal of R . Also, in particular, Ann R ( M ) = (0 M : R M ) is a graded ideal of R .The concept of graded prime ideals and its generalizations have a significant placein graded commutative algebra since they are used in understanding the structureof graded rings. Recall that a proper graded ideal P of R is said to be graded primeideal if x, y ∈ h ( R ) such that xy ∈ P implies x ∈ P or y ∈ P ([ ]). Gradedprime ideals have been extended to graded modules in [ ]. A proper graded R -submodule N of M is said to be a graded prime submodule if whenever x ∈ h ( R )and m ∈ h ( M ) with xm ∈ N , then m ∈ N or x ∈ ( N : R M ). The notionof graded 2-absorbing ideal, which is a generalization of graded prime ideal, wasintroduced in [ ] as the following: a proper graded ideal P of R is said to be graded2-absorbing if whenever x, y, z ∈ h ( R ) such that xyz ∈ P , then either xy ∈ P or xz ∈ P or yz ∈ P . Also, Graded 2-absorbing ideals have been deeply studied in[ ]. Graded 2-absorbing submodules have been introduced and studied in [ ]. Aproper graded R -submodule N of a graded R -module M is said to be graded 2-absorbing if whenever x, y ∈ h ( R ) and m ∈ h ( M ) such that xym ∈ N , then either xm ∈ N or ym ∈ N or xy ∈ ( N : R M ). Graded 2-absorbing submodules have beengeneralized into graded n -absorbing submodules in [ ]. Actually, the concept ofgraded 2-absorbing submodule is a generalization of graded prime submodules.Recently, in [ ], a new class of graded ideals has been introduced, which isan intermediate class of graded ideals between graded prime ideals and graded 2-absorbing ideals. A proper graded ideal P of R is said to be a graded 1-absorbingprime ideal if for each non-units x, y, z ∈ h ( R ) with xyz ∈ P , then either xy ∈ P or z ∈ P . Clearly, every graded prime ideal of R is graded 1-absorbing prime andevery graded 1-absorbing prime ideal of R is graded 2-absorbing. The next exampleshows that not every graded 2-absorbing ideal of R is graded 1-absorbing prime. Example . [ ] Consider R = Z + 3 x Z [ x ] and G = Z . Then R is G -graded by R = Z , R j = 3 Z x j for j ≥ and R j = { } otherwise. Now, P = 3 x Z [ x ] is a gradedideal of R , so by Lemma 1.1, P is a graded ideal of R . By ( [ ] , Example 2.2), P is a -absorbing ideal of R , and then it is graded -absorbing. On the other hand, P is not graded -absorbing prime as , , x ∈ h ( R ) such that (3)(9)(3 x ) ∈ P while (3)(9) / ∈ P and x / ∈ P . Thus we have the following chain:
N GRADED 1-ABSORBING PRIME SUBMODULES 3 graded prime ideals ⇛ graded -absorbing prime ideals ⇛ graded -absorbingideals .On the other hand, we have another chain: graded prime submodules ⇛ graded -absorbing submodules .Thus we realize that there is a missing part in the second chain, which is betweengraded prime submodules and graded 2-absorbing submodules. Then we define themissing part of the chain as graded 1-absorbing prime submodules. In this article,we are motivated by [ ] to introduce and study the concept of graded 1-absorbingprime submodule. A proper graded R -submodule N of M is said to be a graded1-absorbing prime R -submodule of M if for all non-unit elements x, y ∈ h ( R ) and m ∈ h ( M ) with xym ∈ N , either xy ∈ ( N : R M ) or m ∈ N . We show that thenew concept is a generalization of graded prime submodules at the same time it isa special graded 2-absorbing submodule. Several properties of a graded 1-absorbingprime submodule have been obtained. We investigate graded 1-absorbing primesubmodules when the components { M g : g ∈ G } are multiplication R e -modules.
2. Graded -Absorbing Prime Submodules In this section, we introduce and study the concept of graded 1-absorbing primesubmodules.
Definition . Let M be a graded R -module and N be a proper graded R -submodule of M . Then N is said to be a graded -absorbing prime R -submodule of M if for all non-unit elements x, y ∈ h ( R ) and m ∈ h ( M ) with xym ∈ N , either xy ∈ ( N : R M ) or m ∈ N . Example . Let R be a graded local ring with graded maximal ideal X satisfies X = { } . If M is a graded R -module, then every proper graded R -submoduleof M is graded -absorbing prime. To prove that, let N be a proper graded R -submodule of M , x, y ∈ h ( R ) be non-units and m ∈ h ( M ) such that xym ∈ N .Since xy ∈ X = { } , xy ∈ ( N : R M ) , and hence N is a graded -absorbing prime R -submodule of M . Proposition . Let M be a graded R -module and N be a graded R -submoduleof M . (1) If N is a graded prime R -submodule of M , then N is a graded -absorbingprime R -submodule of M . (2) If N is a graded -absorbing prime R -submodule of M , then N is a graded -absorbing R -submodule of M . Proof. (1) Let x, y ∈ h ( R ) be non-units and m ∈ h ( M ) such that xym ∈ N . Then z = xy ∈ h ( R ) with zm ∈ N . Since N is graded prime, either z = xy ∈ ( N : R M ) or m ∈ N . Hence, N is a graded 1-absorbing prime R -submodule of M .(2) Let x, y ∈ h ( R ) and m ∈ h ( M ) such that xym ∈ N . If x is unit, then ym ∈ N . If y is unit, then xm ∈ N . Suppose that x and y are non-units.Since N is graded 1-absorbing prime, either xy ∈ ( N : R M ) or m ∈ N , asneeded. Hence, N is a graded 2-absorbing R -submodule of M . (cid:3) The next example shows that the converse of Proposition 2.3 (1) is not true ingeneral.
AHMAD
KA’ABNEH
AND RASHID
ABU-DAWWAS
Example . Consider R = Z , M = Z [ x ] and G = Z . Then R is G -gradedby R = Z and R j = { } otherwise. Also, M is G -graded by M j = Z x j for j ≥ and M j = { } otherwise. As x ∈ h ( M ) , N = h x i is a graded R -submodule of M .By Example 2.2, N is a graded -absorbing prime R -submodule of M . On the otherhand, N is not graded prime R -submodule of M . The next example shows that the converse of Proposition 2.3 (2) is not true ingeneral.
Example . Consider R = Z , M = Z [ i ] and G = Z . Then R is G -gradedby R = Z and R = { } . Also, M is G graded by M = Z and M = i Z . Then N = h i is a graded -absorbing R -submodule of M . On the other hand, ∈ h ( R ) is non-unit and ∈ h ( M ) with (2)(2)(3) ∈ N , but (2)(2) / ∈ ( N : R M ) and / ∈ N .Hence, N is not graded -absorbing prime R -submodule of M . So, we have the following chain: graded prime submodules ⇛ graded -absorbing prime submodules ⇛ graded -absorbing submodules .And the converse of each implication is not true in general. Proposition . Let M be a graded R -module and N be a graded R -submoduleof M . If N is a graded -absorbing prime R -submodule of M , then ( N : R M ) is agraded -absorbing prime ideal of R . Proof.
Let x, y, z ∈ h ( R ) be non-units such that xyz ∈ ( N : R M ). Assumethat m ∈ M . Then xyzm g ∈ N for all g ∈ G . Since N is graded 1-absorbingprime, either xy ∈ ( N : R M ) or zm g ∈ N for all g ∈ G . If zm g ∈ N for all g ∈ G ,then zm = z X g ∈ G m g ! = X g ∈ G zm g ∈ N , which implies that z ∈ ( N : R M ). Hence,( N : R M ) is a graded 1-absorbing prime ideal of R . (cid:3) Similarly, one can prove the following:
Proposition . Let M be a graded R -module and N be a graded R -submoduleof M . If N is a graded -absorbing prime R -submodule of M , then ( N : R m ) is agraded -absorbing prime ideal of R for all m ∈ h ( M ) − N . The next example shows that the converse of Proposition 2.6 is not true ingeneral.
Example . Consider R = Z , M = Z × Z and G = Z . Then R is G -graded by R = Z and R = { } . Also, M is G -graded by M = Z × { } and M = { } × Z . As (3 , ∈ h ( M ) , N = h (3 , i is a graded R -submodule of M such that ( N : R M ) = { } is a graded prime ideal of R , so ( N : R M ) is a graded -absorbing prime ideal of R . On the other hand, , ∈ h ( R ) are non-units and (1 , ∈ h ( M ) with (3)(2)(1 , ∈ N , but (3)(2) / ∈ ( N : R M ) and (1 , / ∈ N . Hence, N is not graded -absorbing prime R -submodule of M . Lemma . Let M be a graded R -module and N be a graded -absorbing prime R -submodule of M . For a graded R -submodule L of M and for non-unit elements x, y ∈ h ( R ) , if xyL ⊆ N , then either xy ∈ ( N : R M ) or L ⊆ N . Proof.
Assume that L * N . Then there is an element 0 = m ∈ L − N , andthen there exists g ∈ G such that m g / ∈ N . Note that, m g ∈ L as L is a graded N GRADED 1-ABSORBING PRIME SUBMODULES 5 R -submodule. By assumption, we have xym g ∈ N . Since N is a graded 1-absorbingprime R -submodule, either xy ∈ ( N : R M ) or m g ∈ N . The second choice implies acontradiction, we conclude that xy ∈ ( N : R M ), as desired. (cid:3) Theorem . Let M be a graded R -module and N be a proper graded R -submodule of M . Then N is a graded -absorbing prime R -submodule of M if andonly if whenever I , J are proper graded ideals of R and L is a graded R -submoduleof M such that IJ L ⊆ N , then either IJ ⊆ ( N : R M ) or L ⊆ N . Proof.
Suppose that N is a graded 1-absorbing prime R -submodule of M .Assume that I , J are proper graded ideals of R and L is a graded R -submodule of M such that IJ L ⊆ N . Suppose that IJ * ( N : R M ). Then there exist x ∈ I and y ∈ J such that xy / ∈ ( N : R M ), and then there exist g, h ∈ G such that x g y h / ∈ ( N : R M ). Note that, x g ∈ I and y h ∈ J as I and J are graded ideals.Since I and J are proper, x g and y h are non-units. Now, x g y h L ⊆ N , so by Lemma2.9, L ⊆ N , as needed. Conversely, let x, y ∈ h ( R ) be non-units and m ∈ h ( M )such that xym ∈ N . Then I = h x i , J = h y i are proper graded ideals of R and L = h m i is a graded R -submodule of M such that IJ L ⊆ N . By assumption, either IJ ⊆ ( N : R M ) or L ⊆ N . If IJ ⊆ ( N : R M ), then xy ∈ ( N : R M ). Hence, N is agraded 1-absorbing prime R -submodule of M . (cid:3) Let M be a G -graded R -module, N be a graded R -submodule of M and g ∈ G such that N g = M g . In [ ], N is said to be a g -prime R -submodule of M if whenever r ∈ R e and m ∈ M g such that rm ∈ N , then either r ∈ ( N : R M ) or m ∈ N .Also, in [ ], A graded ideal P of R is said to be a g -1-absorbing prime ideal of R if P g = R g and whenever non-unit elements x, y, z in R g such that xyz ∈ P , then xy ∈ P or z ∈ P . We introduce the following definition: Definition . Let M be a G -graded R -module, N be a graded R -submoduleof M and g ∈ G such that N g = M g . Then N is said to be a g - -absorbing prime R -submodule of M if for all non-unit elements x, y ∈ R e and m ∈ M g with xym ∈ N ,either xy ∈ ( N : R M ) or m ∈ N . Remark . Clearly, every g -prime R -submodule is g - -absorbing prime. How-ever, the converse is not true in general; N = h x i in Example 2.4 is a graded -absorbing prime R -submodule, so it is a g - -absorbing prime R -submodule for all g ∈ G = Z , but if we choose g = 1 , then N will be not graded g -prime R -submodule.Now, we are going to prove that if R e is not local ring, then a graded R -submodulewill be an e - -absorbing prime if and only if it is an e -prime R -submodule. Lemma . Let M be a graded R -module. If M has an e - -absorbing prime R -submodule that is not an e -prime R -submodule, then the sum of every non-unitelement of R e and every unit element of R e is a unit element of R e . Proof.
Let N be an e -1-absorbing prime R -submodule of M that is not an e -prime R -submodule. Then there exist a non-unit r ∈ R e and m ∈ M g such that rm ∈ N but r / ∈ ( N : R M ) and m / ∈ N . Choose a non-unit element a ∈ R e . Then wehave that ram ∈ N and m / ∈ N . Since N is e -1-absorbing prime, ra ∈ ( N : R M ). Let u ∈ R e be a unit element. Assume that a + u is non-unit. Then r ( a + u ) m ∈ N . As N is e -1-absorbing prime, r ( a + u ) ∈ ( N : R M ). This means that ru ∈ ( N : R M ), i.e., r ∈ ( N : R M ), which is a contradiction. Thus, we have a + u is a unit element. (cid:3) Theorem . Let M be a graded R -module. If M has an e - -absorbing prime R -submodule that is not an e -prime R -submodule, then R e is a local ring. AHMAD
KA’ABNEH
AND RASHID
ABU-DAWWAS
Proof.
By Lemma 2.13, the sum of every non-unit element of R e and everyunit element of R e is a unit element of R e , and then by ([ ], Lemma 1), R e is a localring. (cid:3) Corollary . Let R be a graded ring such that R e is not local ring. Supposethat M is a graded R -module. Then a graded R -submodule N of M is an e - -absorbing prime R -submodule if and only if N is an e -prime R -submodule of M . Proposition . Let { N k } k ∈ ∆ be a chain of graded -absorbing prime R -submodules of M . Then \ k ∈ ∆ N k is a graded -absorbing prime R -submodule of M . Proof.
Let x, y ∈ h ( R ) be non-units and m ∈ h ( M ) such that xym ∈ \ k ∈ ∆ N k .Suppose that m / ∈ \ k ∈ ∆ N k . Then m / ∈ N i for some i ∈ ∆. Since xym ∈ N i and N i is graded 1-absorbing prime, xy ∈ ( N i : R M ). For any k ∈ ∆, if N i ⊆ N k ,then xy ∈ ( N k : R M ), and then xy ∈ \ k ∈ ∆ N k : R M ! . If N k ⊆ N i , then m / ∈ N k .Since xym ∈ N k and N k is graded 1-absorbing prime, xy ∈ ( N k : R M ), and then xy ∈ \ k ∈ ∆ N k : R M ! . Hence, \ k ∈ ∆ N k is a graded 1-absorbing prime R -submoduleof M . (cid:3) Lemma . Let M be a graded R -module. If M is finitely generated, thenthe union of every chain of proper graded R -submodules of M is a proper graded R -submodule of M . Proof.
Suppose that M = h x , x , ..., x n i for some x , ..., x n ∈ M . Let { N k } k ∈ ∆ be a chain of proper graded R -submodules of M . If [ k ∈ ∆ N k = M , then x , x ∈ [ k ∈ ∆ N k , and then x ∈ N j and x ∈ N r for some j, r ∈ ∆. If N j ⊆ N r , then x , x ∈ N r . If N r ⊆ N j , then x , x ∈ N j . In fact, this works for all x i ’s. So,there exists i ∈ ∆ such that N i contains all the generators x , ..., x n of M , and then N i = M , which is a contradiction. (cid:3) Proposition . Let { N k } k ∈ ∆ be a chain of graded -absorbing prime R -submodules of M . If M is finitely generated, then [ k ∈ ∆ N k is a graded -absorbingprime R -submodule of M . Proof.
By Lemma 2.17, [ k ∈ ∆ N k is a proper graded R -submodule of M . Let x, y ∈ h ( R ) be non-units and m ∈ h ( M ) such that xym ∈ [ k ∈ ∆ N k . Suppose that m / ∈ [ k ∈ ∆ N k . Then m / ∈ N k for all k ∈ ∆. Now, since xym ∈ [ k ∈ ∆ N k , xym ∈ N i for some i ∈ ∆, and since N i is graded 1-absorbing prime, xy ∈ ( N i : R M ) ⊆ N GRADED 1-ABSORBING PRIME SUBMODULES 7 [ k ∈ ∆ N k : R M ! . Hence, [ k ∈ ∆ N k is a graded 1-absorbing prime R -submodule of M . (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 . Lemma . ( [ ] , Lemma 4.8) Suppose that f : M → S is a graded R -homomorphism of 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 K is a graded -absorbing prime R -submodule of S and f − ( K ) = M ,then f − ( K ) is a graded -absorbing prime R -submodule of M . (2) Assume that f is a graded epimorphism. If L is a graded -absorbing prime R -submodule of M containing Ker ( f ) , then f ( L ) is a graded -absorbingprime R -submodule of S . Proof. (1) By Lemma 2.19, f − ( K ) is a graded R -submodule of M . Let x, y ∈ h ( R ) be non-units and m ∈ h ( M ) such that xym ∈ f − ( K ). Then f ( m ) ∈ h ( S ) such that xyf ( m ) = f ( xym ) ∈ K . Since K is graded 1-absorbing prime, either xy ∈ ( K : R S ) or f ( m ) ∈ K , and then either xy ∈ ( f − ( K ) : R M ) or m ∈ f − ( K ). Hence, f − ( K ) is a graded 1-absorbing prime R -submodule of M .(2) By Lemma 2.20, f ( L ) is a graded R -submodule of S . Let x, y ∈ h ( R )be non-units and s ∈ h ( S ) such that xys ∈ f ( L ). Since f is gradedepimorphism, there exists m ∈ h ( M ) such that s = f ( m ), and then f ( xym ) = xyf ( m ) = xys ∈ f ( L ), which implies that f ( xym ) = f ( t ) forsome t ∈ L , and then xym − t ∈ Ker ( f ) ⊆ L , which gives that xym ∈ L .Since L is graded 1-absorbing prime, either xy ∈ ( L : R M ) or m ∈ L , andthen either xy ∈ ( f ( L ) : R S ) or s = f ( m ) ∈ f ( L ). Hence, f ( L ) is a graded1-absorbing prime R -submodule of S . (cid:3) If M is a G -graded R -module and L is a graded R -submodule of M , then M/L is a G -graded R -module by ( M/L ) g = ( M g + L ) /L for all g ∈ G ([ ]). Corollary . Let M be a graded R -module and L ⊆ N be graded R -submodules of M . If N is a graded -absorbing prime R -submodule of M , then N/L is a graded -absorbing prime R -submodule of M/L . Proof.
Define f : M → M/L by f ( x ) = x + L . Then f is a graded epimorphismwith Ker ( f ) = L ⊆ N . So, by Proposition 2.21, f ( N ) = N/L is a graded 1-absorbing prime R -submodule of M/L . (cid:3) Definition . Let M be a graded R -module, L be a proper graded R -submoduleof M and N be a graded -absorbing prime R -submodule of M with L ⊆ N . Then N is said to a minimal graded -absorbing prime R -submodule with respect to L ifthere is no a graded -absorbing prime R -submodule P of M such that L ⊆ P ⊂ N . AHMAD
KA’ABNEH
AND RASHID
ABU-DAWWAS
Theorem . Let M be a graded R -module and L be a proper graded R -submodule of M . If N is a graded -absorbing prime R -submodule of M such that L ⊆ N , then there exists a minimal graded -absorbing prime R -submodule withrespect to L that it is contained in N . Proof.
Let X be the set of all graded 1-absorbing prime R -submodules N i of M such that L ⊆ N i ⊆ N . Then X is non-empty as it contains N . Consider ( X, ⊇ ).Let { N k } k ∈ ∆ be a chain in X . Then by Proposition 2.16, \ k ∈ ∆ N k is a graded 1-absorbing prime R -submodule of M , and then by Zorn’s Lemma, X has a maximalelement K . So, K is a graded 1-absorbing prime R -submodule of M such that L ⊆ K ⊆ N . If K is not minimal graded 1-absorbing prime R -submodule withrespect to L , then there exists a graded 1-absorbing prime R -submodule P of M such that L ⊆ P ⊆ K , and then P ∈ X , which implies that K ⊆ P , and hence P = K . Thus, K is a minimal graded 1-absorbing prime R -submodule with respectto L . (cid:3) Let I be a graded ideal of R . Then Grad ( I ) is the intersection of all gradedprime ideals of R containing I ([ ]). Similarly, if N is a graded R -submodule of M , then the graded radical of N is Grad ( N ) which is the intersection of all gradedprime R -submodules of M containing N . Motivated by this, we have the followingdefinition. Definition . (1) Let I be a graded ideal of R . Then the -graded rad-ical of I is the intersection of all graded -absorbing prime ideals of R containing I , and is denoted by Grad ( I ) . If I = R or R has no graded -absorbing prime ideals, we define Grad ( I ) = R . (2) Let N be a graded R -submodule of M . Then the -graded radical of N is theintersection of all graded -absorbing prime R -submodules of M containing N , and is denoted by Grad ( N ) . If N = M or M has no graded -absorbingprime R -submodules, we define Grad ( N ) = M . Remark . Since every graded prime ideal is graded -absorbing prime, then Grad ( I ) ⊆ Grad ( I ) for all graded ideal I of R . Similarly, Grad ( N ) ⊆ Grad ( N ) for all graded R -submodule N of M . Motivated by ([ ], Proposition 1.2), we state the following proposition, and theproof is elementary. Proposition . Let M be a graded R -module and N, K be two graded R -submodules of M . Then the following statements hold: (1) N ⊆ Grad ( N ) . (2) Grad ( Grad ( N )) ⊆ Grad ( N ) . (3) Grad ( N T K ) ⊆ Grad ( N ) T Grad ( K ) . Proposition . Let M be a graded R -module and N be a graded R -submoduleof M . Then Grad (( N : R M )) ⊆ ( Grad ( N ) : R M ) . Proof. If Grad ( N ) = M , then it is done. Suppose that Grad ( N ) = M . Thenthere exists a graded 1-absorbing prime R -submodule L of M such that N ⊆ L , andthen by Proposition 2.6, ( L : R M ) is a graded 1-absorbing prime ideal of R such that( N : R M ) ⊆ ( L : R M ), which implies that Grad (( N : R M )) ⊆ ( L : R M ), and hence Grad (( N : R M )) M ⊆ L . In fact, in similar way, Grad (( N : R M )) M ⊆ L i for every N GRADED 1-ABSORBING PRIME SUBMODULES 9 graded 1-absorbing prime R -submodule L i of M such that N ⊆ L i . So, Grad (( N : R M )) M ⊆ Grad ( N ), which means that Grad (( N : R M )) ⊆ ( Grad ( N ) : R M ). (cid:3) Proposition . Let R be a graded ring and I be a graded ideal of R . If I isa graded -absorbing prime ideal of R , then I e is a -absorbing prime ideal of R e . Proof.
Let x, y, z ∈ R e be non-units such that xyz ∈ I e . Then x, y, z ∈ h ( R )such that xyz ∈ I . Since I is graded 1-absorbing prime, either xy ∈ I or z ∈ I . If xy ∈ I , then xy ∈ R e R e T I ⊆ R e T I = I e . If z ∈ I , then z ∈ R e T I = I e . Hence, I e is a 1-absorbing prime ideal of R e . (cid:3) Proposition . Let M be a G -graded R -module and N be a graded R -submodule of M . If N is a graded -absorbing prime R -submodule of M , then N g is a -absorbing prime R e -submodule of M g for al g ∈ G . Proof.
Let g ∈ G . Suppose that x, y ∈ R e be non-units and m ∈ M g suchthat xym ∈ N g . Then x, y ∈ h ( R ) and m ∈ h ( M ) such that xym ∈ N . Since N is graded 1-absorbing prime, either xy ∈ ( N : R M ) or m ∈ N . If xy ∈ ( N : R M ),then xy ∈ R e such that xyM g ⊆ xyM ⊆ N , also, xyM g ⊆ R e M g ⊆ M g , and hence xyM g ⊆ N T M g = N g , which implies that xy ∈ ( N g : R e M g ). If m ∈ N , then m ∈ N T M g = N g . Hence, N g is a 1-absorbing prime R e -submodule of M g . (cid:3) Let M be a G -graded R -module and g ∈ G . In the sense of [ ], M g is said to be amultiplication R e -module if whenever N is an R e -submodule of M g , then N = IM g for some ideal I of R e . Note that, since I ⊆ ( N : R e M g ), N = IM g ⊆ ( N : R e M g ) M g ⊆ N . So, if M g is a multiplication R e -module, then N = ( N : R e M g ) M g forevery R e -submodule N of M g . Proposition . Let M be a G -graded R -module and g ∈ G such that M g isa faithful multiplication R e -module. Suppose that I is a -absorbing prime ideal of R e . Then whenever x, y ∈ R e are non-units and m ∈ M g such xym ∈ IM g , theneither xy ∈ I or m ∈ IM g . Proof.
Apply ([ ], Theorem 5) on the R e -module M g . (cid:3) Theorem . Let M be a G -graded R -module and g ∈ G such that M g is afaithful multiplication R e -module. If I is a graded -absorbing prime ideal of R and ( IM ) g = M g , then IM is a g - -absorbing prime R -submodule of M . Proof.
By Lemma 1.1 (3), IM is a graded R -submodule of M . Since I is agraded 1-absorbing prime ideal of R , by Proposition 2.29, I e is a 1-absorbing primeideal of R e . Let x, y ∈ R e be non-units and m ∈ M g such that xym ∈ IM . Then xym = x e y e m g = ( xym ) g ∈ ( IM ) g = I e M g . By Proposition 2.31, either xy ∈ I e or m ∈ I e M g ⊆ IM . If xy ∈ I e , then xy ∈ I and then xyM ⊆ IM , which implies that xy ∈ ( IM : R M ). Hence, IM is a g -1-absorbing prime R -submodule of M . (cid:3) Proposition . Let M be a graded R -module and N be a graded -absorbingprime R/Ann R ( M ) -submodule of M as M is a graded R/Ann R ( M ) -module. If U ( R/Ann R ( M )) = { r + Ann R ( M ) : r ∈ U ( R ) } , then N is a graded -absorbingprime R -submodule of M as M is a graded R -module. Proof.
Let x, y ∈ h ( R ) be non-units and m ∈ h ( M ) such that xym ∈ N .Then x + Ann R ( M ) , y + Ann R ( M ) ∈ h ( R/Ann R ( M )) are non-units such that( x + Ann R ( M ))( y + Ann R ( M )) m = xym + Ann R ( M ) m ∈ N . Since N is agraded 1-absorbing prime R/Ann R ( M )-submodule of M , either xy + Ann R ( M ) ∈ KA’ABNEH
AND RASHID
ABU-DAWWAS ( N : R/Ann R ( M ) M ) or m ∈ N . If xy + Ann R ( M ) ∈ ( N : R/Ann R ( M ) M ), then xyM ⊆ N , which implies that xy ∈ ( N : R M ). Hence, N is a graded 1-absorbingprime R -submodule of M as M is a graded R -module. (cid:3) Theorem . Let M be a G -graded R -module and g ∈ G such that M g is amultiplication R e -module and U ( R e /Ann R e ( M g )) = { r + Ann R e ( M g ) : r ∈ U ( R e ) } .Assume that I is a graded -absorbing prime ideal of R containing Ann R ( M ) . Then ( I e /Ann R e ( M g )) M g is a g - -absorbing prime R -submodule of M . Proof.
Since M g is a multiplication R e -module, M g is a faithful multiplication R e /Ann R e ( M g )-module (see [ ], page 759). Also, since I is a graded 1-absorbingprime ideal of R , I e is a 1-absorbing prime ideal of R e by Proposition 2.29, and then I e /Ann R e ( M g ) is a 1-absorbing prime ideal of R e /Ann R e ( M g ) by ([ ], Proposition7). So, ( I e /Ann R e ( M g )) M g is a g -1-absorbing prime R e /Ann R e ( M g )-submodule of M by Theorem 2.32, which implies that ( I e /Ann R e ( M g )) M g is a g -1-absorbing prime R -submodule of M by ([ ], Proposition 8). (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] R. Abu-Dawwas, ¨U. Tekir, S. Ko¸c and E. Yıldız, Graded 1-absorbing prime ideals, accepted inthe Sao Paulo Journal of Mathematical Sciences, in press.[3] K. Al-Zoubi and R. Abu-Dawwas, On graded 2-absorbing and weakly graded 2-absorbing sub-modules, Journal of Mathematical Sciences: Advances and Applications, 28 (2014), 45-60.[4] 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.[5] K. Al-Zoubi and F. Qarqaz, An intersection condition for graded prime ideals, Bollettinodell’Unione Matematica Italiana, 11 (2018), 483-488.[6] S. E. Atani, On graded prime submodules, Chiang Mai Journal of Science, 33 (1) (2006), 3-7.[7] S. E. Atani and F. E. K. Saraei, Graded modules which satisfy the gr-radical formula, ThaiJournal of Mathematics, 8 (1) (2010), 161-170.[8] A. Badawi and E. Y. Celikel, On 1-absorbing primary ideals of commutative rings, Journal ofAlgebra and Its Applications, 19 (6), 2050111 (2020)[9] Z. A. El-Bast and P. F. Smith, Multiplication modules, Communications in Algebra, 16 (1988),755-779.[10] F. Farzalipour, P. Ghiasvand, On the union of graded prime submodules, Thai Journal ofMathematics, 9 (1) (2011), 49-55.[11] M. Hamoda and A. E. Ashour, On graded n -ansorbing submodules, Le Matematiche, (2015),243-254, doi:10.4418/2015.70.2.16.[12] R. Hazrat, Graded rings and graded Grothendieck groups, Cambridge University press, 2016.[13] S. R. Naghani and H. F. Moghimi, On graded 2-absorbing and graded weakly 2-absorbingideals of a commutative ring, C¸ ankaya University Journal of Science and Engineering, 13 (2)(2016), 11-17.[14] C. Nastasescu and F. Oystaeyen, Methods of graded rings, Lecture Notes in Mathematics,1836, Springer-Verlag, Berlin, 2004.[15] M. Refai and K. Al-Zoubi, On graded primary ideals, Turkish Journal of Mathematics, 28(2004), 217-229.[16] M. Refai, M. Hailat and S. Obiedat, Graded radicals and graded prime spectra, Far EastJournal of Mathematical Sciences, (2000), 59-73.[17] E. A. Ugurlu, On 1-absorbing prime submodules, (2020), arXiv:2007.01103v1.[18] A. Yassine, M. J. Nikmehr, R. Nikandish, On 1-absorbing prime ideals of commutative rings,Journal of Algebra and its Applications, (2020), doi:10.1142/S0219498821501759. N GRADED 1-ABSORBING PRIME SUBMODULES 11
Department of Mathematics, University of Jordan, Amman, Jordan
Email address : [email protected] Department of Mathematics, Yarmouk University, Irbid, Jordan
Email address ::