aa r X i v : . [ m a t h . A C ] J a n ON GRADED STRONGLY -ABSORBING PRIMARY IDEALS RASHID ABU-DAWWAS
Abstract.
Let G be a group with identity e and R be a G -graded commutativering with nonzero unity 1. In this article, we introduce the concept of gradedstrongly 1-absorbing primary ideals. A proper graded ideal P of R is said to be agraded strongly 1-absorbing primary ideal of R if whenever nonunit homogeneouselements x, y, z ∈ R such that xyz ∈ P , then either xy ∈ P or z ∈ Grad ( { } ) (thegraded radical of { } ). Several properties of graded strongly 1-absorbing primaryideals are investigated. Many results are given to disclose the relations betweenthis new concept and others that already exist. Namely, the graded prime ideals,the graded primary ideals, and the graded 1-absorbing primary ideals. Introduction
Throughout this article, all rings are commutative with a nonzero unity 1. Let G be a group with identity e . Then a ring 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 be written uniquely as X g ∈ G x g , where x g is the component of x in R g . Also, weset h ( R ) = [ g ∈ G R g . Moreover, it has been proved in [5] that R e is a subring of R and1 ∈ R e . Let I be an ideal of a graded ring R . Then I is said to be a 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 ofa graded ring need not be graded (see [5]).Let I be a proper graded ideal of R . Then the graded radical of I is Grad ( I ) = ( x = X g ∈ G x g ∈ R : for all g ∈ G, there exists n g ∈ N such that x n g g ∈ I ) . Note that Grad ( I ) is always a graded ideal of R (see [6]). Proposition 1.1. ( [4] ) Let R be a G -graded ring. (1) If I and J are graded ideals of R , then I + J , IJ and I T J are graded idealsof R . (2) If a ∈ h ( R ) , then Ra is a graded ideal of R . Recall that a proper graded ideal Q of R is said to be graded primary if whenever x, y ∈ h ( R ) with xy ∈ Q , then x ∈ Q or y ∈ Grad ( Q ). In this case P = Grad ( Q ) isa graded prime ideal of R , and Q is said to be graded P -primary. In [2], Al-Zoubiand Sharafat introduced a generalization of graded primary ideals called graded2-absorbing primary ideals. A proper graded ideal I of R is called a graded 2-absorbing primary ideal of R if whenever x, y, z ∈ h ( R ) and xyz ∈ I , then xy ∈ I or Mathematics Subject Classification.
Key words and phrases.
Graded primary ideals, graded 1-absorbing primary ideals. xz ∈ Grad ( I ) or yz ∈ Grad ( I ). The concept of graded 2-absorbing primary idealsis generalized in many ways (see for example [8]). Recently in [3], we consider anew class of graded ideals called the class of graded 1-absorbing primary ideals. Aproper graded ideal P of R is said to be a graded 1-absorbing primary ideal of R if whenever nonunit elements x, y, z ∈ h ( R ) such that xyz ∈ P , then xy ∈ P or z ∈ Grad ( P ). Clearly, every graded primary ideal is graded 1-absorbing primaryideal. The next example shows that the converse is not true in general. Example 1.2. [3]
Let K be a field and assume that R = K [ X, Y ] is Z -gradedwith degX = 1 = degY . Consider the graded ideal P = ( X , XY ) of R . Then Grad ( P ) = ( X ) . Since for X.Y.X ∈ P , either X.Y ∈ P or X ∈ Grad ( P ) , P is agraded -absorbing primary ideal of R . On the other hand, P is not graded primaryideal of R by ( [7] , Example 2.11). Also, it is clear that every graded 1-absorbing primary ideal is graded 2-absorbingprimary ideal. The next example shows that the converse is not true in general.
Example 1.3. [3]
Let R = Z [ i ] and G = Z . Then R is G -graded by R = Z and R = i Z . Consider P = 12 R . Then as ∈ h ( R ) , P is a graded ideal of R . By( [2] , Example 2.2 (ii)), P is a graded -absorbing primary ideal of R . On the otherhand, , ∈ h ( R ) such that . . ∈ P , but neither . ∈ P nor ∈ Grad ( P ) . So, P is not graded -absorbing primary ideal of R . In this article, we follow [1] to introduce and study a subclass of the class of graded1-absorbing primary ideals that does not necessarily contain all graded primaryideals. A proper graded ideal P of R is called graded strongly 1-absorbing primaryif whenever nonunit elements x, y, z ∈ h ( R ) and xyz ∈ P , then xy ∈ P or z ∈ Grad ( { } ). Several properties of graded strongly 1-absorbing primary ideals areinvestigated. Many results are given to disclose the relations between this newconcept and others that already exist. Namely, the graded prime ideals, the gradedprimary ideals, and the graded 1-absorbing primary ideals.2. Graded Strongly -Absorbing Primary Ideals In this section, we introduce and study the concept of graded strongly 1-absorbingprimary ideals.
Definition 2.1.
Let R be a graded ring and P be a proper graded ideal of R . Then P is said to be a graded strongly -absorbing primary ideal of R if whenever x, y, z ∈ h ( R ) such that xyz ∈ P , then either xy ∈ P or z ∈ Grad ( { } ) . Clearly, every graded strongly 1-absorbing primary ideal is a graded 1-absorbingprimary ideal. However, the next example shows that the converse is not true ingeneral. The same example shows that a graded prime ideal (in particular a gradedprimary ideal) is not necessarily a graded strongly 1-absorbing primary ideal.
Example 2.2.
Let R = Z [ i ] and G = Z . Then R is G -graded by R = Z and R = i Z . Consider P = 3 R . Then as ∈ h ( R ) , P is a graded ideal of R . It is clearthat P is graded prime, and then graded -absorbing primary. However, P is notgraded strongly 1-absorbing primary. Indeed, , ∈ h ( R ) with . . ∈ P but neither . ∈ P nor ∈ Grad ( { } ) = { } . A graded ring is said to be graded local if it has one graded maximal ideal only.The next theorem gives a characterization of graded strongly 1-absorbing primaryideals.
N GRADED STRONGLY 1-ABSORBING PRIMARY IDEALS 3
Theorem 2.3.
Let P be a proper graded ideal of R . Then P is graded strongly -absorbing primary if and only if (1) P is graded -absorbing primary and Grad ( P ) = Grad ( { } ) , or (2) R is graded local with graded maximal ideal X = Grad ( P ) and X ⊆ P .Proof. Suppose that P is graded strongly 1-absorbing primary. Then it is clear that P is graded 1-absorbing primary. Assume that Grad ( P ) = Grad ( { } ). Assumethat there exist nonunit elements a, b ∈ h ( R ) such that ab / ∈ P . Let x ∈ P .Then since P is a graded ideal, x g ∈ P for all g ∈ G , and then abx g ∈ P forall g ∈ G . Since P is graded strongly 1-absorbing primary, x g ∈ Grad ( { } ) forall g ∈ G , which implies that x ∈ Grad ( { } ). So, P ⊆ Grad ( { } ), and hence Grad ( P ) = Grad ( { } ), which is a contradiction. Thus, ab ∈ P for each nonunitelements a, b ∈ h ( R ). Let X be a graded maximal ideal of R . We have that X ⊆ P ,and then X = Grad ( X ) ⊆ Grad ( P ). Thus, X = Grad ( P ) for each graded maximalideal X of R . Hence, we conclude that R is graded local with graded maximal ideal X = Grad ( P ) and X ⊆ P . Conversely, if P is graded 1-absorbing primary and Grad ( P ) = Grad ( { } ), then P is clearly a graded strongly 1-absorbing primary idealof R . Suppose that R is graded local with graded maximal ideal X = Grad ( P ) and X ⊆ P . Then for each nonunit elements a, b ∈ h ( R ) , ab ∈ X ⊆ P , and then P istrivially a graded strongly 1-absorbing primary ideal of R . (cid:3) A graded commutative ring R with unity is said to be a graded domain if R hasno homogeneous zero divisors. A graded commutative ring R with unity is saidto be a graded field if every nonzero homogeneous element of R is unit. The nextexample shows that a graded field need not be a field. Example 2.4.
Let R be a field and suppose that F = { x + uy : x, y ∈ R, u = 1 } .If G = Z , then F is G -graded by F = R and F = uR . Let a ∈ h ( F ) such that a = 0 . If a ∈ F , then a ∈ R and since R is a field, we have a is a unit element.Suppose that a ∈ F . Then a = uy for some y ∈ R . Since a = 0 , we have y = 0 ,and since R is a field, we have y is a unit element, that is zy = 1 for some z ∈ R .Thus, uz ∈ F such that ( uz ) a = uz ( uy ) = u ( zy ) = 1 . , which implies that a is a unit element. Hence, F is a graded field. On the other hand, F is not a fieldsince u ∈ F − { } is not a unit element since (1 + u )(1 − u ) = 0 . Remark 2.5.
Let R be a graded local ring. If P is a graded strongly -absorbingprimary ideal of R , then Grad ( P ) needs not to be graded maximal. To see that, itsuffices to consider any graded local domain which is not graded field. Then { } = Grad ( { } ) is graded strongly -absorbing primary which is not graded maximal. Corollary 2.6.
Let P be a graded prime ideal of R . Then P is graded strongly -absorbing primary if and only if (1) P = Grad ( { } ) , or (2) R is graded local with graded maximal ideal P . Lemma 2.7. [3]
Let R be a graded ring and P be a graded ideal of R . If P is agraded -absorbing primary ideal of R , then Grad ( P ) is a graded prime ideal of R .Proof. Let a, b ∈ h ( R ) such that ab ∈ Grad ( P ). We may assume that a, b arenonunit elements of R . Let k ≥ ab ) k ∈ P .Then k = 2 s for some positive integer s ≥
1. Since ( ab ) k = a k b k = a s a s b k ∈ P and P is a graded 1-absorbing primary ideal of R , we conclude that a s a s = a k ∈ P or R. ABU-DAWWAS b k ∈ P . Hence, a ∈ Grad ( P ) or b ∈ Grad ( P ). Thus Grad ( P ) is a graded primeideal of R . (cid:3) Theorem 2.8.
Let R be a graded ring. Then there exists a graded strongly -absorbing primary ideal of R if and only if (1) Grad ( { } ) is graded prime, or (2) R is graded local.Proof. Suppose that R contains a graded strongly 1-absorbing primary ideal P . If R is not graded local, then by Theorem 2.3, Grad ( P ) = Grad ( { } ). Now, byLemma 2.7, Grad ( P ) is graded prime since P is graded 1-absorbing primary. Thus, Grad ( { } ) is graded prime. Conversely, by Corollary 2.6, if ( R, X ) is graded local,then X is a graded strongly 1-absorbing primary ideal, and if P = Grad ( { } ) isgraded prime, then it is a graded strongly 1-absorbing primary ideal. (cid:3) A graded ring R is said to be graded Artinian if every descending chain of gradedideals of R terminates. Corollary 2.9.
Let n ≥ be an integer and the ring Z /n Z be G -graded by a group G . Then Z /n Z has a graded strongly -absorbing primary ideal if and only if n = p m for some prime p and positive integer m .Proof. By Theorem 2.8, Z /n Z has a graded strongly 1-absorbing primary ideal ifand only if it is graded local or Grad ( { } ) is graded prime. On the other hand, Z /n Z is Artinian and so it is graded Artinian, and then Z /n Z is graded local if andonly it is graded field, that is n is prime number. Now, if n = p k ...p k r r (the primarydecomposition of n ), then Grad ( { } ) = p ...p r Z /n Z . Hence, Grad ( { } ) is gradedprime if and only if r = 1. (cid:3) Let R and S be two G -graded rings. Then R × S is G -graded ring by ( R × S ) g = R g × S g for all g ∈ G . Corollary 2.10.
Let R and S be two G -graded rings. Then R × S has no gradedstrongly -absorbing primary ideal.Proof. The result holds from the fact that R × S is not graded local and Grad ( { R × S } ) = Grad ( { R } ) × Grad ( { S } ) is never graded prime ideal in R × S . (cid:3) Proposition 2.11.
Let P be a proper graded ideal of R . Then P is a graded strongly -absorbing primary ideal of R if and only if whenever IJ K ⊆ P for some propergraded ideals I, J and K of R , then IJ ⊆ P or K ⊆ Grad ( { } ) .Proof. Suppose that P is a graded strongly 1-absorbing primary ideal of R . Assumethat I, J and K be proper graded ideals of R such that IJ K ⊆ P and IJ * P .Then there exist a ∈ I and b ∈ J such that ab / ∈ P . Since a ∈ I and I is a gradedideal, a g ∈ I for all g ∈ G . Similarly, b g ∈ J for all g ∈ G . Since ab / ∈ P , thereexist g, h ∈ G such that a g b h / ∈ P . Let x ∈ K . Then x r ∈ K for all r ∈ G , andthen a g b h x r ∈ P for all r ∈ G . Since P is graded strongly 1-absorbing primary, x r ∈ Grad ( { } ) for all r ∈ G , and then x ∈ Grad ( { } ). Hence, K ⊆ Grad ( { } ).Conversely, let x, y, z ∈ h ( R ) be nonunit elements such that xyz ∈ P . Then I = Rx , J = Ry and K = Rz are proper graded ideals of R with IJ K ⊆ P , and then IJ ⊆ P or K ⊆ Grad ( { } ), which implies that xy ∈ P or z ∈ Grad ( { } ). Hence, P is agraded strongly 1-absorbing primary ideal of R . (cid:3) N GRADED STRONGLY 1-ABSORBING PRIMARY IDEALS 5
Proposition 2.12.
Let R be a graded ring, P and K be two proper graded ideals of R . If P and K are graded strongly -absorbing primary, then so is P T K .Proof. By Proposition 1.1, P T K is a graded ideal of R . Let x, y, z ∈ h ( R ) benonunit elements such that xyz ∈ P T K . Then xyz ∈ P and xyz ∈ K . If z ∈ Grad ( { } ), then it is done. Suppose that z / ∈ Grad ( { } ). Since xyz ∈ P and P is graded strongly 1-absorbing primary, xy ∈ P . Similarly, xy ∈ K . Hence, xy ∈ P T K , and thus P T K is a graded strongly 1-absorbing primary ideal of R . (cid:3) Proposition 2.13.
Let R be a graded ring such that every element of h ( R ) is eithernilpotent or unit. Then Ra is a graded strongly -absorbing primary ideal of R forall nonunit a ∈ h ( R ) .Proof. Let a ∈ h ( R ) be nonunit element. By Proposition 1.1, Ra is a graded idealof R , and as a is nonunit, Ra is proper. Assume that x, y, z ∈ h ( R ) be nonunitelements such that xyz ∈ Ra . Then z ∈ Grad ( { } ), and hence Ra is a gradedstrongly 1-absorbing primary ideal of R . (cid:3) Corollary 2.14.
Let R be a graded ring such that every element of h ( R ) is eithernilpotent or unit. Then every proper graded ideal of R is graded strongly -absorbingprimary.Proof. Let P be a proper graded ideal of R . Assume that x, y, z ∈ h ( R ) be nonunitelements such that xyz ∈ P and z / ∈ Grad ( { } ). Then xyz ∈ h ( R ) is nonunitelement with xyz ∈ R ( xyz ). By Proposition 2.13, R ( xyz ) is graded strongly 1-absorbing primary, and then xy ∈ R ( xyz ) ⊆ P . Hence, P is a graded strongly1-absorbing primary ideal of R . (cid:3) Example 2.15.
Every element of R = Z / Z is either unit or nilpotent. So, if R is G -graded by any group G , then every homogeneous element of R is either unitor nilpotent, and then every proper graded ideal of R is graded strongly -absorbingprimary by Corollary 2.14. Proposition 2.16.
Let R be a graded ring. Then every graded prime ideal of R isgraded strongly -absorbing primary if and only if R is graded local and has at mostone graded prime ideal that is not graded maximal.Proof. Suppose that every graded prime ideal of R is graded strongly 1-absorbingprimary. Assume that R is not graded local and let P be a graded prime ideal of R . Then since P is graded strongly 1-absorbing primary, we get Grad ( { } ) = P byCorollary 2.6. Hence, Grad ( { } ) is the only graded prime ideal of R . Thus, R isgraded local, which is a contradiction. Hence, R is graded local with graded maximalideal X . Now, let P be a graded prime ideal which is not graded maximal. Then Grad ( P ) = Grad ( { } ). Hence, R has at most two graded prime ideals Grad ( { } )and X . Conversely, if every homogeneous element of R is either unit or nilpotent,then it is done by Corollary 2.14. Otherwise, R has exactly two graded prime ideals P ( X . Since Grad ( { } ) is the intersection of graded prime ideals, we get that Grad ( { } ) = P . It is clear that both of Grad ( { } ) and X are graded strongly1-absorbing primary ideals, as desired. (cid:3) Proposition 2.17.
Let R be a graded ring. Then every graded primary ideal of R is graded strongly -absorbing primary if and only if (1) every element of h ( R ) is either nilpotent or unit, or R. ABU-DAWWAS (2) R is graded local with graded maximal ideal X , one graded prime ideal that isnot graded maximal (which is Grad ( { } ) , and every graded X -primary idealcontains X .Proof. Suppose that every graded primary ideal of R is graded strongly 1-absorbingprimary. If (1) does not hold, then by Proposition 2.16, R is graded local with exactlytow graded prime ideals which are Grad ( { } ) and X (the graded maximal ideal).Now, let P be a graded X -primary ideal of R . Then since Grad ( { } ) = Grad ( P ) = X , we have necessarily X ⊆ P (by Theorem 2.3). Conversely, if (1) holds, then theresult follows trivially. Hence, suppose that (2) holds. Let P be a graded primaryideal of R . Then Grad ( P ) = Grad ( { } ) or Grad ( P ) = X . If Grad ( P ) = Grad ( { } ),then by Theorem 2.3, P is graded strongly 1-absorbing primary since every gradedprimary is graded 1-absorbing primary. Now, if Grad ( P ) = X , then also P is gradedstrongly 1-absorbing primary since X ⊆ P . (cid:3) Proposition 2.18.
Let R be a graded ring. Then { } is the only graded strongly -absorbing primary ideal of R if and only if (1) R is a graded field, or (2) R is a graded domain that is not graded local.Proof. Suppose that { } is the only graded strongly 1-absorbing primary ideal of R . If R is graded local with graded maximal ideal X , then X = { } since X is agraded strongly 1-absorbing primary ideal of R . Hence, R is a graded field. Now,if R is not graded local, then Grad ( { } ) = { } is graded prime since Grad ( { } )is a graded strongly 1-absorbing primary ideal. Thus, R is a graded domain. Theconverse is obvious. (cid:3) Lemma 2.19.
Let R be a graded ring. Suppose that P and K are graded ideals of R . Then ( P : K ) = { r ∈ R : rK ⊆ P } is a graded ideal of R .Proof. Clearly, ( P : K ) is an ideal of R . Let r ∈ ( P : K ). Then rK ⊆ P and r = X g ∈ G r g . Let x ∈ K . Then since K is graded, x h ∈ K for all h ∈ G . Now, r g x h ∈ h ( R ) for all g, h ∈ G with X g ∈ G r g x h = X g ∈ G r g ! x h = rx h ∈ rK ⊆ P for all h ∈ G . Since P is graded, r g x h ∈ P for all g, h ∈ G , and then r g x ∈ P for all g ∈ G ,which implies that r g K ⊆ P for all g ∈ G , so r g ∈ ( P : K ) for all g ∈ G . Hence,( P : K ) is a graded ideal of R . (cid:3) Lemma 2.20.
Let P be a graded -absorbing primary ideal of R and K * P be aproper graded ideal of R . Then ( P : K ) is a graded primary ideal of R .Proof. By Lemma 2.19, ( P : K ) is a graded ideal of R and it is proper as K * P .Let x, y ∈ h ( R ) with xy ∈ ( P : K ) and x / ∈ ( P : K ). Clearly, y is a nonunit element.If x is unit, then y ∈ ( P : K ) ⊆ Grad (( P : K )). Hence, we may assume that x and y are nonunit elements of R . Since x / ∈ ( P : K ), there is z ∈ K such that xz / ∈ P ,and then xz g / ∈ P for some g ∈ G . Note that z g ∈ K as K is graded. But xyz g ∈ P and P is graded 1-absorbing primary. Then y ∈ Grad ( P ) ⊆ Grad (( P : K )). Thus,( P : K ) is a graded primary ideal of R . (cid:3) Proposition 2.21.
Let P be a graded strongly -absorbing primary ideal of R and K * Grad ( P ) be a proper graded ideal of R . Then ( P : K ) is a graded strongly -absorbing primary ideal of R . N GRADED STRONGLY 1-ABSORBING PRIMARY IDEALS 7
Proof. If Grad ( P ) = Grad ( { } ), then R is graded local with graded maximal ideal Grad ( P ). In this situation, our assumption K * Grad ( P ) is not satisfied. Thenwe must have that Grad ( P ) = Grad ( { } ) is graded prime. Let a ∈ ( P : K ). Then a g ∈ ( P : K ) for all g ∈ G as ( P : K ) is a graded ideal by Lemma 2.19, and then a g K ⊆ P ⊆ Grad ( P ) for all g ∈ G . Since K * Grad ( P ), we get a g ∈ Grad ( P ) forall g ∈ G , which implies that a ∈ Grad ( P ). Hence, P ⊆ ( P : K ) ⊆ Grad ( P ). Thus, Grad (( P : K )) = Grad ( P ) = Grad ( { } ). Now, the result follows from Theorem 2.3and Lemma 2.20. (cid:3) Let R and S be two G -graded rings. Then a ring homomorphism f : R → S issaid to be graded homomorphism if f ( R g ) ⊆ S g for all g ∈ G . Lemma 2.22.
Suppose that f : R → S is a graded homomorphism. (1) If K is a graded ideal of S , then f − ( K ) is a graded ideal of R . (2) If P is a graded ideal of R with Ker ( f ) ⊆ P , then f ( P ) is a graded ideal of f ( R ) .Proof. (1) Clearly, f − ( K ) is a an ideal of R . Let x ∈ f − ( K ). Then x ∈ R with f ( x ) ∈ K , and x = X g ∈ G x g where x g ∈ R g for all g ∈ G . So, for every g ∈ G , f ( x g ) ∈ f ( R g ) ⊆ S g such that X g ∈ G f ( x g ) = f X g ∈ G x g ! = f ( x ) ∈ K .Since K is graded, f ( x g ) ∈ K for all g ∈ G , i.e., x g ∈ f − ( K ) for all g ∈ G .Hence, f − ( K ) is a graded ideal of R .(2) Clearly, f ( P ) is an ideal of f ( R ). Let y ∈ f ( P ). Then y ∈ f ( R ), and sothere exists x ∈ R such that y = f ( x ). So f ( x ) ∈ f ( P ), which implies that x ∈ P since Ker ( f ) ⊆ P , and hence x g ∈ P for all g ∈ G since P is graded.Thus, y g = ( f ( x )) g = f ( x g ) ∈ f ( P ) for all g ∈ G . Therefore, f ( P ) is agraded ideal of f ( R ). (cid:3) Proposition 2.23.
Suppose that f : R → S is a graded homomorphism. (1) If f is a graded epimorphism and P is a graded strongly -absorbing primaryideal of R containing Ker ( f ) , then f ( P ) is a graded strongly -absorbingprimary ideal of S . (2) If f is a graded monomorphism and K is a graded strongly -absorbing pri-mary ideal of S , then f − ( K ) is a graded strongly -absorbing primary idealof R .Proof. (1) By Lemma 2.22, f ( P ) is a graded ideal of S . Let a, b, c ∈ h ( S ) benonunit elements such that abc ∈ f ( P ). Since f is a graded epimorphism,there exist nonunit elements x, y, z ∈ h ( R ) such that a = f ( x ) , b = f ( y ) and c = f ( z ). Suppose that ab / ∈ f ( P ). Then xy / ∈ P . Hence, since P is a gradedstrongly 1-absorbing primary ideal of R and xyz ∈ P (because Ker ( f ) ⊆ P ),we get that z ∈ Grad ( { R } ). Thus, c = f ( z ) ∈ Grad ( { S } ). Consequently, f ( P ) is a graded strongly 1-absorbing primary ideal of S .(2) By Lemma 2.22, f − ( K ) is a graded ideal of R . Let a, b, c ∈ h ( R ) benonunit elements such that abc ∈ f − ( K ). Suppose that ab / ∈ f − ( K ).Then f ( a ) f ( b ) / ∈ K . Hence, since K is a graded strongly 1-absorbing pri-mary ideal of S and f ( a ) f ( b ) f ( c ) ∈ K , we conclude that f ( c ) ∈ Grad ( { S } ). R. ABU-DAWWAS
Thus, since f is a graded monomorphism, c ∈ Grad ( { R } ). Consequently, f − ( K ) is a graded strongly 1-absorbing primary ideal of R . (cid:3) Let R be a G -graded ring and K be a graded ideal of R . Then R/K is G -gradedby ( R/K ) g = ( R g + K ) /K for all g ∈ G . Lemma 2.24.
Let R be a graded ring and K be a graded ideal of R . If P is a gradedideal of R with K ⊆ P , then P/K is a graded ideal of
R/K .Proof.
Clearly,
P/K is an ideal of
R/K . Let x + K ∈ P/K . Then x ∈ P , and then x g ∈ P for all g ∈ G as P is graded, which implies that x g + K ∈ P/K for all g ∈ G .Hence, P/K is a graded ideal of
R/K . (cid:3) Corollary 2.25.
Let K ⊆ P be two proper graded ideals of a graded ring R . If P is a graded strongly -absorbing primary ideal of R , then P/K is a graded strongly -absorbing primary ideal of R/K .Proof.
Define f : R → R/K by f ( x ) = x + K . Then f is a graded epimorphismwith Ker ( f ) = K ⊆ P . So, by Proposition 2.23, f ( P ) = P/K is a graded strongly1-absorbing primary ideal of
R/K . (cid:3) Corollary 2.26.
Let P be a graded strongly -absorbing primary ideal of R and S be a graded subring of R . Then P T S is a graded strongly -absorbing primary idealof S .Proof. Define f : S → R by f ( x ) = x . Then f is a graded monomorphism. So, byProposition 2.23, f − ( P ) = P T S is a graded strongly 1-absorbing primary ideal of S . (cid:3) Corollary 2.27.
Let P be a graded strongly -absorbing primary ideal of R . Then P e is a strongly -absorbing primary ideal of R e .Proof. Apply Corollary 2.26 with S = R e . (cid:3) Let S be a multiplicatively closed subset of h ( R ). Then S − R is a graded ringwith ( S − R ) g = (cid:8) as , a ∈ R h , s ∈ S ∩ R hg − (cid:9) . Proposition 2.28.
Let R be a graded ring, S be a multiplicatively closed subset of h ( R ) and P is a graded strongly -absorbing primary ideal of R such that P T S = ∅ .Then S − P is a graded strongly -absorbing primary ideal of S − R .Proof. Since P T S = ∅ , S − P is a proper graded ideal of S − R . Let xs ys zs ∈ S − P for some nonunit elements x, y, z ∈ h ( R ) and s , s , s ∈ S . Then there is u ∈ S such that uxyz ∈ P . Suppose that xs ys / ∈ S − P . Then uxy / ∈ P . Hence, z ∈ Grad ( { R } ) since P is a graded strongly 1-absorbing primary ideal of R . Thus, zs ∈ S − Grad ( { R } = Grad ( { S − R } ). (cid:3) Let R be a G -graded ring. Then R [ X ] is G -graded by ( R [ X ]) g = R g [ X ] for all g ∈ G . Lemma 2.29.
Let R be a G -graded ring and P be a graded ideal of R . Then P [ X ] is a graded ideal of R [ X ] .Proof. Let f ( X ) ∈ P [ X ]. Then f ( X ) = a + a X + ... + a n X n for some a , a , ..., a n ∈ P . Since P is a graded ideal, we have ( a k ) g ∈ P for all k = 0 , , ..., n and g ∈ G ,and then ( f ( X )) g = ( a ) g + ( a ) g X + ... + ( a n ) g X n ∈ P [ X ] for all g ∈ G . Hence, P [ X ] is a graded ideal of R [ X ]. (cid:3) N GRADED STRONGLY 1-ABSORBING PRIMARY IDEALS 9
Proposition 2.30.
Let R be a graded ring and P be a proper graded ideal of R . (1) R [ X ] has a graded strongly -absorbing primary ideal if and only if Grad ( { R } ) is a graded prime ideal of R . (2) If P [ X ] is a graded strongly -absorbing primary ideal of R [ X ] , then P is agraded strongly -absorbing primary ideal of R . (3) The graded ideal P + XR [ X ] is never a graded strongly -absorbing primaryideal of R [ X ] . (4) P [ X ] is a graded strongly -absorbing primary ideal of R [ X ] if and only if P [ X ] is graded primary and Grad ( P ) = Grad ( { } ) .Proof. (1) From Theorem 2.8, R [ X ] has a graded strongly 1-absorbing primaryideal if and only if Grad ( { R [ X ] } ) is graded prime (since R [ X ] is never gradedlocal). Moreover, Grad ( { R [ X ] } ) = ( Grad ( { R } ))[ X ] and ( Grad ( { R } ))[ X ]is a graded prime ideal of R [ X ] if and only Grad ( { R } ) is a graded primeideal of R .(2) If P [ X ] is a graded strongly 1-absorbing primary ideal of R [ X ], then byCorollary 2.26, P = P [ X ] T R is a graded strongly 1-absorbing primaryideal of R .(3) Note that X = 1 .X ∈ R e X ⊆ R e [ X ] = ( R [ X ]) e , and so X ∈ h ( R [ X ]),and then by Lemma 1.1, XR [ X ] is a graded ideal of R [ X ], and then alsoby Lemma 1.1, P + XR [ X ] is a graded ideal of R [ X ]. Since R [ X ] is notgraded local and P + XR [ X ] * Grad ( { R [ X ] } ) (because X / ∈ Grad ( { R [ X ] } )), P + XR [ X ] is never a graded strongly 1-absorbing primary ideal of R [ X ].(4) Since R [ X ] is not graded local, P [ X ] is a graded strongly 1-absorbing primaryideal of R [ X ] if and only if P [ X ] is graded Grad ( { R [ X ] } )-primary if andonly if P [ X ] is graded primary and ( Grad ( { R } ))[ X ] = Grad ( { R [ X ] } ) = Grad ( P [ X ]) = ( Grad ( P ))[ X ] if and only if P [ X ] is graded primary and Grad ( { R } ) = Grad ( P ). (cid:3) References [1] F. A. A. Almahdi, E. M. Bouba and A. N. A. Koam, On strongly 1-absorbing primary idealsof commutative rings, Bulletin of the Korean Mathematical Society, 57 (5) (2020), 1205-1213.[2] K. Al-Zoubi and N. Sharafat, On graded 2-absorbing primary and graded weakly 2-absorbingprimary ideals, Journal of Korean Mathematical Society, 54 (2) (2017), 675-684.[3] M. Bataineh and R. Dawwas, Graded 1-absorbing primary ideals, submitted.[4] F. Farzalipour and P. Ghiasvand, On the union of graded prime submodules, Thai Journal ofMathematics, 9 (1) (2011), 49-55.[5] C. Nastasescu and F. V. Oystaeyen, Methods of graded rings, Lecture Notes in Mathematics,1836, Springer-Verlag, Berlin, (2004).[6] M. Refai, M. Hailat and S. Obiedat, Graded radicals and graded prime spectra, Far EastJournal of Mathematical Sciences, (2000), 59-73.[7] F. Soheilnia and A. Y. Darani, On graded 2-absorbing and graded weakly 2-absorbing primaryideals, Kyungpook Mathematical Journal, 57 (4) (2017), 559-580.[8] R. N. Uregen, ¨U. Tekir, K. P. Shum and S. Ko¸c, On graded 2-absorbing quasi primary ideals,Southeast Asian Bulletin of Mathematics, 43 (2019), 601-613.
Rashid Abu-Dawwas, Department of Mathematics, Yarmouk University, Irbid, Jordan.Email address ::