aa r X i v : . [ m a t h . A C ] J a n ON GRADED RADICALLY PRINCIPAL IDEALS
RASHID ABU-DAWWAS
Abstract.
Let R be a commutative G -graded ring with a nonzero unity. In thisarticle, we introduce the concept of graded radically principal ideals. A gradedideal I of R is said to be graded radically principal if Grad ( I ) = Grad ( h c i ) for somehomogeneous c ∈ R , where Grad ( I ) is the graded radical of I . The graded ring R is said to be graded radically principal if every graded ideal of R is graded radicallyprincipal. We study graded radically principal rings. We prove an analogue of theCohen theorem, in the graded case, precisely, a graded ring is graded radicallyprincipal if and only if every graded prime ideal is graded radically principal.Finally we study the graded radically principal property for the polynomial ring R [ X ]. 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]). If I is a graded ideal of R , then R/I isa graded ring with (
R/I ) g = ( R g + I ) /I for all g ∈ G .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 [8]). 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 h a i is a graded ideal of R . The concept of graded principal ideals have been introduced by Ashby in [1]. Agraded ideal I of R is said to be graded principal if I = h c i for some c ∈ h ( R ). Thegraded ring R is said to be graded principal if every graded ideal of R is graded Mathematics Subject Classification.
Key words and phrases.
Graded radical ideals, graded principal ideals, graded radically principalideals, graded radically principal rings. principal. In this article, we follow [3] to introduce the concept of graded radicallyprincipal ideals. A graded ideal I of R is said to be graded radically principal if Grad ( I ) = Grad ( h c i ) for some c ∈ h ( R ). The graded ring R is said to be gradedradically principal if every graded ideal of R is graded radically principal. Westudy graded rings with this property. Clearly, every graded principal ring is gradedradically principal, we prove that the converse is not true in general.Graded prime ideals have been introduced and studied by Refai, Hailat andObiedat in [8]. A proper graded ideal P of R is said to be graded prime if whenever x, y ∈ h ( R ) such that xy ∈ P , then either x ∈ P or y ∈ P . We prove the Cohen-type theorem for graded radically principal property, that is, a graded ring R isgraded radically principal if and only if every graded prime ideal is graded radicallyprincipal.Atani and Tekir in [2] introduced the avoidance graded prime theorem, that is, if P ⊆ P S ... S P n , where P and P i ’s are graded prime ideals, then P ⊆ P i for some i , this property does not hold for infinite set of graded ideals P i ’s. We characterizegraded rings with avoidance property for infinite set of graded prime ideals. Finallywe study the graded radically principal property for the polynomial ring R [ X ].2. Graded Radically Principal Ideals
In this section, we introduce and study graded radically principal ideals.
Definition 2.1.
Let R be a graded ring and I be a graded ideal of R . Then I issaid to be graded radically principal if Grad ( I ) = Grad ( h c i ) for some c ∈ h ( R ) . Thegraded ring R is said to be graded radically principal if every graded ideal of R isgraded radically principal. Clearly, every graded principal ring is graded radically principal. However, thenext example shows that the converse is not true in general.
Example 2.2.
Let K be a field, S = K [ X, Y ] and G = Z . Then S is G -graded by S n = M i + j = n,i,j ≥ Kx i y j with deg ( x ) = deg ( y ) = 1 . Then I = h X , XY, Y i is a gradedideal of S . Suppose that R = S/I = K [ X, Y ] /I = K [ x, y ] where x = X and y = Y .Then R is G -graded by R n = ( S/I ) n = ( S n + I ) /I . Note that, if P is a graded primeideal of R , then x = y = 0 ∈ P , and then x, y ∈ P , and hence h x, y i ⊆ P . Since h x, y i is a graded maximal ideal of R , P = h x, y i . Thus, the only graded prime idealof R is P = h x, y i = Grad ( h i ) . Now, let I be a graded ideal of R . Then either I = R = Grad ( h i ) or I ⊆ P , and in this case, Grad ( I ) = P = Grad ( h i ) . Hence, R is graded radically principal. On the other hand, R is not graded principal sincethe graded ideal P = h x, y i is not graded principal. Proposition 2.3.
Let R be a graded ring and I be a graded radically principal idealof R . Then there exists c ∈ I T h ( R ) such that I = Grad ( h c i ) .Proof. Since I is graded radically principal, there exists a ∈ h ( R ) such that Grad ( I ) = Grad ( h a i ), and then a ∈ h a i ⊆ Grad ( h a i ) = Grad ( I ), which implies that a n ∈ I for some positive integer n . Now, a ∈ h ( R ), so a ∈ R g for some g ∈ G , and then a n = a.a...a | {z } n − times ∈ R g R g ...R g | {z } n − times ⊆ R g n ⊆ h ( R ). Thus, c = a n ∈ I T h ( R ) such that I ⊆ Grad ( I ) = Grad ( h a i ) = Grad ( h a n i ) = Grad ( h c i ) ⊆ I . (cid:3) N GRADED RADICALLY PRINCIPAL IDEALS 3
Proposition 2.4.
Let R be a graded ring. Suppose that I and J are two gradedradically principal ideals of R . Then IJ and I T J are graded radically principalideals of R .Proof. By Proposition 1.1, IJ and I T J are graded ideals of R . Also, by Proposition2.3, Grad ( I ) = Grad ( h x i ) and Grad ( J ) = Grad ( h y i ) for some x ∈ I T h ( R ) and y ∈ J T h ( R ). So, xy ∈ h ( R ) such that Grad ( IJ ) = Grad ( I T J ) = Grad ( I ) T Grad ( J ) = Grad ( h x i ) T Grad ( h y i ) = Grad ( h xy i ). Hence, IJ and I T J are graded radicallyprincipal ideals of R . (cid:3) Proposition 2.5.
Let R be a graded radically principal ring. Then R/I is a gradedradically principal ring for every graded ideal I of R .Proof. Let
J/I be a graded ideal of
R/I . Then J is a graded ideal of R , and then J = Grad ( h c i ) for some c ∈ h ( R ). So, c ∈ h ( R/I ) such that
Grad ( J/I ) =
Grad ( h c i ).Hence, R/I is a graded radically principal ring. (cid:3)
The next theorem gives an analogue of the Cohen-type theorem for graded radi-cally principal rings.
Theorem 2.6.
Let R be a graded ring. Then R is graded radically principal if andonly if every graded prime ideal of R is graded radically principal.Proof. Suppose that every graded prime ideal of R is a graded radically principalideal. Assume that X is the set of all graded ideals of R that are not graded radicallyprincipal. We show that X = ∅ . Suppose that X = ∅ . Let I ⊆ I ⊆ ... ⊆ I n ⊆ ... be an increasing chain in X . Suppose that I = [ i I i . Then I is a graded ideal of R . If I is graded radically principal, then Grad ( I ) = Grad ( h c i ) for some c ∈ h ( R ).Since c ∈ Grad ( I ), c k ∈ I for some positive integer k , and then c k ∈ I j for some j , which implies that c ∈ Grad ( I j ), and hence Grad ( I j ) = Grad ( h c i ), which is acontradiction. Thus, I ∈ E and clearly, I i ⊆ I for all i . By Zorn’s lemma, X hasa maximal element, say P . We show that P is graded prime. Let x, y ∈ h ( R ) suchthat xy ∈ P . Suppose that x, y / ∈ P , and let P = P + h x i and P = P + h y i . Since P is maximal in X with P $ P and P $ P , P and P are graded radically principalideals of R , and then by Proposition 2.4, P P is a graded radically principal idealof R , but Grad ( P P ) = Grad ( P ) since P P ⊆ P and P ⊆ P P . This gives acontradiction. Hence, P is a graded prime ideal of R which is not graded radicallyprincipal, that is a contradiction. Thus, X = ∅ . The converse is clear. (cid:3) Let S ⊆ h ( R ) be a multiplicative set. Then S − R is a graded ring with ( S − R ) g = (cid:8) as , a ∈ R h , s ∈ S ∩ R hg − (cid:9) . Corollary 2.7.
Let R be a graded radically principal ring. If S is a multiplicativesubset of h ( R ) , then S − R is a graded radically principal ring.Proof. Let P be a graded prime ideal of S − R . Then P = S − K for some gradedprime ideal of R . Since R is graded radically principal, K = Grad ( h c i ) for some c ∈ h ( R ), and since Grad ( P ) = S − K = S − Grad ( h c i ) = Grad ( S − h c i ), we havethat P is a graded radically principal ideal of S − R . Hence, by Theorem 2.6, S − R is a graded radically principal ring. (cid:3) Let R and R be two G -graded rings. Then R × R is G -graded ring by ( R × R ) g = ( R ) g × ( R ) g for all g ∈ G . R. ABU-DAWWAS
Corollary 2.8.
Let R , R ,..., R n be G -graded rings and R = R × R × ... × R n .Then R is graded radically principal if and only if R i is graded radically principalfor all ≤ i ≤ n .Proof. Suppose that R is graded radically principal. Then by Proposition 2.5, R i = R/I i , where I i = R × R × ... × R i − ×{ }× R i +1 × ... × R n , is graded radically principalfor all 1 ≤ i ≤ n . Conversely, let P be a graded prime ideal of R . Then there existsa graded prime ideal K j of R j for some j such that P = R × ... × K j × ... × R n .Since R j is graded radically principal, there exists c j ∈ K j T h ( R j ) such that K j = Grad ( h c j i ), and then c = (1 , ..., c j , ..., ∈ h ( R ) such that Grad ( P ) = Grad ( h c i ).So, P is a graded radically principal ideal of R , and hence by Theorem 2.6, R is agraded radically principal ring. (cid:3) Theorem 2.9.
Let R be a graded ring. Then R is a graded radically principal ringif and only if for every graded prime ideal P of R , we have P * [ P * K K .Proof. Suppose that R is a graded radically principal ring. Let P be a graded primeideal of R . Then P = Grad ( h c i ) for some c ∈ h ( R ). If K is a graded prime idealof R such that P * K , then c / ∈ K , and then c / ∈ [ P * K K . Hence, P * [ P * K K .Conversely, let P be a graded prime ideal of R . Since P * [ P * K K , there exists c ∈ P such that c / ∈ K whenever P * K , and then c g / ∈ K for some g ∈ G . Notethat c g ∈ P as P is a graded ideal, and clearly, Grad ( h c g i ) ⊆ P . If K is a gradedprime ideal of R containing c g , then P ⊆ K . Thus, P ⊆ \ c g ∈ K K = Grad ( h c g i ).Hence, P = Grad ( h c g i ). By Theorem 2.6, R is graded radically principal. (cid:3) For a graded ring R , it is well known that if P is a graded prime ideal of R such that P ⊆ [ i ∈ ∆ P i , where P i ’s are graded prime ideals of R and ∆ is finite, then P ⊆ P i for some i ∈ ∆. This result does not hold for an infinite set ∆. The followingresult characterizes graded rings with this property, and the proof is immediate fromTheorem 2.9. Corollary 2.10.
Let R be a graded ring. Then R is a graded radically principalring if and only if R has the graded avoidance property, that is, if P ⊆ [ i ∈ ∆ P i , where P and P i ’s are graded prime ideals of R , then P ⊆ P i for some i ∈ ∆ . Corollary 2.11.
Let R be a graded ring. If R has finitely many graded prime ideals,then R is a graded radically principal ring.Proof. If R has finitely many graded prime ideals, then the graded avoidance prop-erty holds, and then by Corollary 2.10, it follows that R is graded radically princi-pal. (cid:3) A graded commutative ring R with unity is said to be a graded integral domainif R has no homogeneous zero divisors. A graded commutative ring R with unity issaid to be a graded field if every nonzero homogeneous element of R is unit. Thenext example shows that a graded field need not be a field. N GRADED RADICALLY PRINCIPAL IDEALS 5
Example 2.12.
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 . Proposition 2.13.
Let R be a graded integral domain. Then R is a graded field ifand only if R [ X ] is a graded radically principal ring.Proof. Suppose that R is a graded field. Then R [ X ] is a graded principal domain,which implies that R is a graded radically principal ring. Conversely, let a ∈ h ( R ) −{ } . Suppose that I is the graded ideal of R [ X ] generated by a and X . Since R [ X ] isgraded radically principal, Grad ( I ) = Grad ( h f i ) for some homogeneous f ∈ I . Since a ∈ I , a ∈ Grad ( h f i ), and then there exists a positive integer n such that a n = gf forsome g ∈ R [ X ]. Since R is a graded integral domain, 0 = deg ( a n ) = deg ( g )+ deg ( f ),which implies that f is a nonzero constant. Also, since X ∈ I , X ∈ Grad ( h f i ), andthen there exists a positive integer m such that X m = hf for some h ∈ R [ X ]. Since f is constant, 1 = b m f , where b m is the coefficient of X m in h , which implies that f isa unit element, and then I = R [ X ], so there exist s, t ∈ R [ X ] such that 1 = sa + tX ,and then 1 = s (0) a , where s (0) ∈ R ⊆ h ( R [ X ]). Hence, a is a unit element. So, R is a graded field. (cid:3) Example 2.14.
Let K be a field, R = K [ X ] and G = Z . Then R is G -graded by R j = KX j for j ≥ , and R j = 0 otherwise. Since K is a field, K is a graded field,and then K [ X ] is graded radically principal by Proposition 2.13. Example 2.15.
Let K be a field, R = K [ X ] and G = Z . Then R is G -gradedby R = h , x , x , ... i , R = h x, x , x , ... i and R = h x , x , x , ... i . By Proposition2.13, K [ X ] is graded radically principal. First strongly graded rings have been introduced and studied in [6], a G -gradedring R is said to be first strong if 1 ∈ R g R g − for all g ∈ supp ( R, G ), where supp ( R, G ) = { g ∈ G : R g = { }} . In fact, it has been proved that R is first strongly G -graded if and only if supp ( R, G ) is a subgroup of G and R g R h = R gh for all g, h ∈ supp ( R, G ). We introduce the following:
Proposition 2.16.
Every G -graded field is first strongly graded.Proof. Let R be a G -graded field. Suppose that g ∈ supp ( R, G ). Then R g = { } ,and then there exists 0 = x ∈ R g . Since R is a graded field, we conclude that thereexists y ∈ h ( R ) such that xy = 1. Since y ∈ h ( R ), y ∈ R h for some h ∈ G , and then1 = xy ∈ R g R h ⊆ R gh . So, 0 = 1 ∈ R gh T R e , which implies that gh = e , that is h = g − . Hence, 1 = xy ∈ R g R g − , and thus R is first strongly graded. (cid:3) Corollary 2.17.
Let R be a graded integral domain. If R [ X ] is a graded radicallyprincipal ring, then R is first strongly graded.Proof. Apply Proposition 2.13 and Proposition 2.16. (cid:3)
Theorem 2.18.
Let R be a graded ring. If R [ X ] is graded radically principal, then R is graded radically principal and every graded prime ideal of R is graded maximal. R. ABU-DAWWAS
Proof.
Since R [ X ] is graded radically principal, by Proposition 2.5, R [ X ] / h X i isgraded radically principal, and then R is graded radically principal. Let P be agraded prime ideal of R . Then since ( R/P ) [ X ] = R [ X ] /P [ X ] is graded radicallyprincipal by Proposition 2.5, and then R/P is a graded field by Proposition 2.13.Hence, P is a graded maximal ideal of R . (cid:3) References [1] W. T. Ashby, On graded principal ideal domains, Journal of Algebra, Number Theory andApplications 24 (2) (2012), 159-171.[2] S. E. Atani and ¨U Tekir, On the graded primary avoidance theorem, Chiang Mai Journal ofScience, 34 (2) (2007), 161-164.[3] M. Aqalmoun and M. El ouarrachi, Radically principal rings, Khayyam Journal of Mathematics,6 (2) (2020), 243-249.[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, Various types of strongly graded rings, Abhath Al-Yarmouk Journal (Pure Sciencesand Engineering Series) 4 (2) (1995), 9-19.[7] M. Refai and R. Abu-Dawwas, On Generalizations of Graded Second Submodules, ProyeccionesJournal of Mathematics, 39 (6) (2020), 1547-1564.[8] M. Refai, M. Hailat and S. Obiedat, Graded radicals and graded prime spectra, Far EastJournal of Mathematical Sciences, (2000), 59-73.
Rashid Abu-Dawwas, Department of Mathematics, Yarmouk University, Irbid, Jordan.Email address ::