Graded Weakly prime Ideals of Non-Commutative Rings
aa r X i v : . [ m a t h . R A ] J a n GRADED WEAKLY PRIME IDEALS OF NON-COMMUTATIVERINGS
AZZH SAAD ALSHEHRY
AND
RASHID ABU-DAWWAS
Abstract.
In this article, we consider the structure of graded rings, not neces-sarily commutative nor with unity, and study the graded weakly prime ideals. Weinvestigate the graded rings in which all graded ideals are graded weakly prime.Several properties are given, and several examples to support given propositionsare constructed. We initiate the study of graded weakly total prime ideals andinvestigate graded rings for which every proper graded ideal is graded weakly totalprime. Introduction
Throughout this article, all rings are not necessarily commutative nor with unity.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 ; here R g R h denotes the additive subgroup of R consisting of all finite sums ofelements r g s h with r g ∈ R g and s h ∈ R h . We denote this by G ( R ). It is clear that R g is an R e -module for all g ∈ G . The elements of R g are called homogeneous ofdegree g . If x ∈ R , then x can be written as X g ∈ G x g , where x g is the component of x in R g . Also, we set h ( R ) = [ g ∈ G R g . Moreover, it has been proved in [3] that R e is asubring of R and if R has a unity 1, then 1 ∈ 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 . The following example shows that an ideal of a gradedring need not be graded. Example 1.1. ( [1] ) Consider R = M ( K ) (the ring of all × matrices with entriesfrom a field K ) and G = Z (the group of integers modulo ). Then R is G -gradedby R = (cid:18) K K (cid:19) , R = (cid:18) KK (cid:19) and R = R = { } .Consider the ideal I = h (cid:18) (cid:19) i of R . Note that, (cid:18) (cid:19) ∈ I such that (cid:18) (cid:19) = (cid:18) (cid:19)| {z } ∈ R + (cid:18) (cid:19)| {z } ∈ R . If I is a graded ideal of R , then (cid:18) (cid:19) ∈ I which is a contradiction. So, I is not graded ideal of R . Mathematics Subject Classification.
Key words and phrases.
Graded weakly prime ideals, graded weakly total prime ideals, gradedtotal prime ideals, graded prime ideals.
Lemma 1.2. ( [1] ) Let R be a graded ring. (1) If I and J are graded ideals of R , then I + J and I T J are graded ideals of R , (2) If x ∈ h ( R ) , then Rx is a graded ideal of R . Graded prime ideals of graded commutative rings with unity have been introducedand studied in [4]. A proper graded ideal P of a graded commutative ring R withunity is said to be graded prime if whenever x, y ∈ h ( R ) such that xy ∈ P , theneither x ∈ P or y ∈ P . The concept of graded prime ideals and its generalizationshave an outstanding location in graded commutative algebra. They are valuabletools to determine the properties of graded commutative rings. Various general-izations of graded prime ideals have been studied. Indeed, Atani introduced in [2]the concept of graded weakly prime ideals. A proper graded ideal P of a gradedcommutative ring R with unity is said to be a graded weakly prime ideal of R ifwhenever x, y ∈ h ( R ) such that 0 = xy ∈ P , then x ∈ P or y ∈ P . In [1], gradedprime ideals of graded non-commutative rings have been studied. In this article, weconsider the structure of graded rings, not necessarily commutative nor with unity,and study the graded weakly prime ideals.By Theorem 2.12 of Atani [2], the following statements are equivalent for a gradedideal P of G ( R ) with P = R where R is a graded commutative ring with unity:(1) P is a graded weakly prime ideal of G ( R ).(2) For each g, h ∈ G , the inclusion 0 = IJ ⊆ P with R e -submodules I of R g and J of R h implies that I ⊆ P or J ⊆ P .For graded rings that are not necessarily commutative, it is clear that (2) doesnot imply (1). So, we define a graded ideal of G ( R ) to be graded weakly prime asfollows: a graded ideal P of G ( R ) with P = R is said to be a graded weakly primeideal of G ( R ) if for each g, h ∈ G , the inclusion 0 = IJ ⊆ P with R e -submodules I of R g and J of R h implies that I ⊆ P or J ⊆ P . In [1], the standard definitionof a graded prime ideal P for a graded noncommutative ring R is that P = R andwhenever I and J are graded ideals of R such that IJ ⊆ P , then either I ⊆ P or J ⊆ P . Accordingly, we define a graded ideal of a graded ring R to be gradedweakly prime as follows: a proper graded ideal P of R is said to be a graded weaklyprime ideal of R if whenever I and J are graded ideals of R such that 0 = IJ ⊆ P ,then either I ⊆ P or J ⊆ P .Note that by definition, a graded weakly prime ideal is a proper graded ideal of agraded ring. It is therefore not possible that every graded ideal of a graded ring is agraded weakly prime ideal. However, a graded ring whose zero ideal is graded primeis called a graded prime ring. In this sense, every graded ring is a graded weaklyprime ring since the zero ideal is always graded weakly prime. We may therefore saythat every graded ideal of a graded ring is graded weakly prime when every propergraded ideal of the graded ring is a graded weakly prime ideal. In this article, weinvestigate the graded rings in which all graded ideals are graded weakly prime.In [1], a proper graded ideal P of a graded ring R is a graded total prime idealif xy ∈ P implies x ∈ P or y ∈ P for x, y ∈ h ( R ). In this article, we introduce theconcept of graded weakly total prime ideals. A proper graded ideal P of a gradedring R is a graded weakly total prime ideal if 0 = xy ∈ P implies x ∈ P or y ∈ P for x, y ∈ h ( R ). We investigate graded rings for which every proper graded ideal isgraded weakly total prime. RADED WEAKLY PRIME IDEALS OF NON-COMMUTATIVE RINGS 3 Graded Weakly Prime Ideals
In this section, we study the structure of graded rings, not necessarily commuta-tive nor with unity, in which every graded ideal is graded weakly prime.
Definition 2.1. (1)
A proper graded ideal P of R is said to be a graded weaklyprime ideal of R if whenever I and J are graded ideals of R such that = IJ ⊆ P , then either I ⊆ P or J ⊆ P . (2) A graded ideal of G ( R ) with P = R is said to be a graded weakly prime idealof G ( R ) if for each g, h ∈ G , the inclusion = IJ ⊆ P with R e -submodules I of R g and J of R h implies that I ⊆ P or J ⊆ P . Our early proposition is Proposition 2.2 of Atani [2] in a further general scene.
Proposition 2.2.
Let P be a graded weakly prime ideal of R . If P is not gradedprime ideal of R , then P = 0 .Proof. Since P is graded weakly prime ideal of R which is not prime, there existgraded ideals I * P and J * P with 0 = IJ ⊆ P . If P = 0, then 0 = P ⊆ ( I + P )( J + P ) ⊆ P , which implies that either I ⊆ P or J ⊆ P , which is acontradiction. Hence, P = 0. (cid:3) Proposition 2.3.
Let R be a graded ring with unity and P be a graded ideal of G ( R ) with P = R . Then P is a graded weakly prime ideal of G ( R ) if and only ifwhenever x, y ∈ h ( R ) such that = xRy ⊆ P , then either x ∈ P or y ∈ P .Proof. Suppose that P is a graded weakly prime ideal of G ( R ). Let x, y ∈ h ( R )such that 0 = xRy ⊆ P . Then x ∈ R g and y ∈ R h for some g, h ∈ G , and then R e x is an R e -submodule of R g and R e y is an R e -submodule of R h . Since R has unity,0 = ( R e x )( R e y ) ⊆ P , whence x ∈ R e x ⊆ P or y ∈ R e y ⊆ P . Conversely, assumethat g, h ∈ G . Suppose that IJ ⊆ P for R e -submodule I of R g and R e -submodule J of R h , where I * P and J * P . Let x ∈ I − P , y ∈ J − P , z ∈ I T P , and w ∈ J T P . Since x + z, y + w / ∈ P , we should have 0 = ( x + z ) R ( y + w ). Consideringall combinations where z and/or w equal zero shows that 0 = xy = xw = zy = zw ,and hence IJ = 0. (cid:3) We are interested in the structure of graded rings in which every graded idealis graded weakly prime. Note that by definition, a graded weakly prime ideal is aproper graded ideal of a graded ring. It is therefore not possible that every gradedideal of a graded ring is a graded weakly prime ideal. However, a graded ring whosezero ideal is graded prime is called a graded prime ring. In this sense, every gradedring is a graded weakly prime ring since the zero ideal is always graded weaklyprime. We may therefore say that every graded ideal of a graded ring is gradedweakly prime when every proper graded ideal of the graded ring is a graded weaklyprime ideal. If R = 0, then it is clear that every graded ideal of R is graded weaklyprime. In particular, if a graded ideal P of a graded ring R is graded weakly primebut not a graded prime ideal, then every graded ideal of P as a graded ring is gradedweakly prime by Proposition 2.2. Proposition 2.4.
Every graded ideal of a graded ring R is graded weakly prime ifand only if for any graded ideals P and Q of R , P Q = P , P Q = Q , or P Q = 0 .Proof.
Suppose that every graded ideal of R is graded weakly prime. Let P, Q begraded ideals of R . If P Q = R , then P Q is graded weakly prime. If 0 = P Q ⊆ P Q ,then we have P ⊆ P Q or Q ⊆ P Q , that is, P = P Q or Q = P Q . If
P Q = R , then A. S. ALSHEHRY AND R. DAWWAS we have P = Q = R whence R = R . Conversely, let K be a proper graded ideal of R and suppose that 0 = P Q ⊆ K for graded ideals P and Q of R . Then we haveeither P = P Q ⊆ K or Q = P Q ⊆ K . (cid:3) Corollary 2.5.
Let R be a graded ring in which every graded ideal of R is gradedweakly prime. Then for any graded ideal P of R , either P = P or P = 0 . The next example shows that the converse of Corollary 2.5 is not true in general.
Example 2.6.
Let K be a field and R = K L K L K . Then every ideal of R is idempotent. So, if R is G -graded by any group G , then every graded ideal of R is idempotent. On the other hand, if we consider the trivial graduation for R by agroup G , then P = K L L will be a graded ideal of R which is not graded weaklyprime. Suppose that a graded ring R with unity has a graded maximal ideal X and X = 0. Thus, the product of any two graded ideals contained in X is zero. It isobvious that every proper graded ideal of R is contained in X , and for any gradedideal P , P R = RP = P . Hence, every graded ideal of R is graded weakly prime. Inthis case, note that, X is the only graded prime ideal of R . In particular, Corollary2.5 yields that if a graded ring R has the property that every graded ideal is gradedweakly prime, then either R = R , or R = 0. Note that, R is neither 0 nor R inthe next example. Example 2.7.
Let S be a ring such that S = 0 , and let K be a field. Suppose that R = K L S L S and G = Z . Then R is G -graded by R = K L L and R =0 L S L S . Now, X = R is a graded maximal ideal of R and X = 0 . Considerthe ideal P = K L L S of R . To prove that P is graded ideal, let x ∈ P . Then x = a +0+ s for some a ∈ K and s ∈ S , and then x = ( a +0+0)+(0+0+ s ) = x + x where x = a + 0 + 0 ∈ R T P and x = 0 + 0 + s ∈ R T P . Hence, P is a gradedideal of R . Similarly, I = K L S L is a graded ideal of R , and = I ⊆ P but I * P , which means that P is not a graded weakly prime ideal of R . If a graded ring R satisfying R = R has a graded maximal ideal X and X = 0,then every proper graded ideal of R is contained in X . However, it is possible that XR = X . Thus, such a graded ring does not necessarily have the property thatevery graded ideal is graded weakly prime, see the following example. Example 2.8.
Let K be a field, S = K K K K and G = Z . Then S is G -graded ring by S = K K
00 0 0 and S = K
00 0 K . Now, S hasa unique graded maximal ideal L = x y z : x, y, z ∈ K . Consider thegraded ideal P = y : y ∈ K of S . Then R = S/P is a graded ringby R j = ( S j + P ) /P for j = 0 , . Note that, R = R and X = L/P is a gradedmaximal ideal of R whose square is zero, the proper graded ideals RX and XR arenot graded weakly prime. RADED WEAKLY PRIME IDEALS OF NON-COMMUTATIVE RINGS 5
Proposition 2.9.
Let R be a graded ring such that every graded ideal of R is gradedweakly prime and R = R . Then R has at most two graded maximal ideals.Proof. Suppose that R has three distinct graded maximal ideals, say X , X , and X . Then X X = 0. Thus we have 0 = X X ⊆ X T X . But this implies either X ⊆ X or X ⊆ X , a contradiction. (cid:3) The next example shows that the condition R = R in Proposition 2.9 is necessary. Example 2.10.
Let X be the unique maximal ideal of Z . Then R = X L X L X is a ring trivially graded by a group G whose all graded ideals are graded weaklyprime and having more than two graded maximal ideals. Proposition 2.11.
Let R be a graded ring such that every graded ideal of R isgraded weakly prime. If R has two graded maximal ideals, then their product is zero.Moreover, if R has a unity, then R is a direct sum of two graded simple rings.Proof. Suppose that R has two distinct graded maximal ideals X and X . Thensince X T X is graded weakly prime and X X ⊆ X T X , we have X X =0, and similarly X X = 0. If R has a unity, then X T X = ( X T X ) R =( X T X ) ( X + X ) ⊆ X X + X X = 0, which implies that R ∼ = R/X L R/X . (cid:3) Remark 2.12.
Suppose that every graded ideal of a graded ring R is graded weaklyprime. By Proposition 2.2 and Corollary 2.5, any nontrivial idempotent graded idealof R is a graded prime ideal. Recall that the intersection of all graded prime idealsof a graded ring R is called the graded prime radical of R . We denote the gradedprime radical of R by GP ( R ) , and the sum of all graded ideals whose square is zeroby GN ( R ) . Theorem 2.13.
Let R be a graded ring such that every graded ideal of R is gradedweakly prime and R = R . Then GP ( R ) = GN ( R ) and ( GP ( R )) = ( GN ( R )) = 0 .Proof. Let a, b ∈ GN ( R ). Then there exist finitely many square zero graded ideals I , I , ..., I k such that a, b ∈ I + I + ... + I k . Since I j = 0 for each j , ( I + I + ... + I k ) m = 0 for some m , but then ( I + I + ... + I k ) = 0 by Corollary 2.5. Hence,( GN ( R )) = 0. This implies that if P is any graded prime ideal of R , GN ( R ) ⊆ P and consequently, GN ( R ) ⊆ GP ( R ). We should note that R contains at least onegraded prime ideal. Indeed, if R contains a nonzero idempotent graded ideal, thenby Remark 2.12, it should be graded prime. If every graded ideal is nilpotent, thensince R = R , GN ( R ) = R is a graded prime ideal. If GP ( R ) is not a graded primeideal, then ( GP ( R )) = 0. This implies that GP ( R ) ⊆ GN ( R ) by the definitionof GN ( R ), and hence the result follows. Suppose that GP ( R ) is a graded primeideal. In this case, we will show that GN ( R ) should also be graded prime. Thisimplies that GP ( R ) ⊆ GN ( R ) by the definition of GP ( R ), and hence the resultfollows. Suppose that IJ ⊆ GN ( R ) for graded ideals I and J of R . Since GN ( R )is graded weakly prime, we have J ⊆ GN ( R ) or I ⊆ GN ( R ) provided that IJ = 0.Suppose that IJ = 0. If I = 0 or J = 0, then J ⊆ GN ( R ) or I ⊆ GN ( R ).If both I and J are not square zero, then they are graded prime ideals, but then eitherI = I ⊆ IJ = 0 or J = J ⊆ IJ = 0, a contradiction. Thus GN ( R ) is agraded prime ideal, and hence the result follows. (cid:3) Corollary 2.14.
Let R be a graded ring such that every graded ideal of R is gradedweakly prime. Then every nonzero graded ideal of R/GN ( R ) is graded prime. A. S. ALSHEHRY AND R. DAWWAS
Corollary 2.15.
Let R be a graded ring such that every graded ideal of R is gradedweakly prime. Then ( GN ( R )) = 0 and every graded prime ideal contains GN ( R ) .In fact, there are three possibilities: (1) GN ( R ) = R . (2) GN ( R ) = GP ( R ) is the smallest graded prime ideal and all other gradedprime ideals are idempotent. If GN ( R ) = 0 , then it is the only non-idempotentgraded prime ideal. (3) GN ( R ) = GP ( R ) is not a graded prime ideal.Proof. If R = 0, then GN ( R ) = R . So clearly ( GN ( R )) = 0, and there are nograded prime ideals. If R = R , then by Theorem 2.13, GP ( R ) = GN ( R ) and( GP ( R )) = ( GN ( R )) = 0. By the definition of GP ( R ), every graded prime idealcontains GN ( R ) = GP ( R ). If GN ( R ) = GP ( R ) is graded prime, it is evidently thesmallest graded prime ideal and all other graded prime ideals are idempotent. (cid:3) Question 2.16.
Let R be a graded ring such that every graded ideal of R is gradedweakly prime. If GN ( R ) = GP ( R ) is not a graded prime ideal, what could weconclude about GN ( R ) ?. Assume that M is a left 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 called homogeneous of degree g . It is clear that M g isan R e -submodule of M for all g ∈ G . We assume that 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 . It isknown that an R -submodule of a graded R -module need not be graded.Let M be a left R -module. The idealization R (+) M = { ( r, m ) : r ∈ R and m ∈ M } of M is a ring with componentwise addition and multiplication; ( x, m ) + ( y, m ) =( x + y, m + m ) and ( x, m )( y, m ) = ( xy, xm + ym ) for each x, y ∈ R and m , m ∈ M . Let G be an abelian group and M be a G -graded R -module. Then X = R (+) M is G -graded by X g = R g (+) M g for all g ∈ G . Note that, X g is an additive sub-group of X for all g ∈ G . Also, for g, h ∈ G , X g X h = ( R g (+) M g )( R h (+) M h ) =( R g R h , R g M h + R h M g ) ⊆ ( R gh , M gh + M hg ) ⊆ ( R gh , M gh ) = X gh as G is abelian [5]. Lemma 2.17.
Let G be an abelian group, M be a G -graded R -module, P be anideal of R and N be an R -submodule of M such that P M ⊆ N . Then P (+) N is agraded ideal of R (+) M if and only if P is a graded ideal of R and N is a graded R -submodule of M .Proof. Follows from [5, Proposition 3.3]. (cid:3)
Example 2.18.
Let G be an abelian group and M be a G -graded R -module. Con-sider the G -graded ring R (+) M whose graded ideals are precisely of the form P (+) N where P is a graded ideal of R and N is a graded R -submodule of M containing P M . (1) Let R be a graded prime ring that contains exactly one nonzero proper gradedideal P . Then every graded ideal of S = R (+) P is graded weakly prime: thegraded maximum ideal P = P (+) P is idempotent and the nonzero minimalgraded ideal P = 0(+) P is nilpotent, both of which are graded prime. RADED WEAKLY PRIME IDEALS OF NON-COMMUTATIVE RINGS 7 (2)
Every graded ideal of S = S (+) P is graded weakly prime: The gradedmaximum ideal Q = P (+) P is idempotent and the three nonzero nilpotentgraded ideals are Q = P (+) P , Q = 0(+) P , and Q = P (+)0 . We introduce an example that gives a nonzero idempotent graded weakly primeright ideal that is not graded prime, and this is unlike the case of graded weaklyprime two sided ideals.
Example 2.19.
Let K be a field, consider the ring R = (cid:18) K K K (cid:19) and G = Z .Then R is G -graded by R = (cid:18) K K (cid:19) , R = (cid:18) K (cid:19) and R = R = 0 . Thenthe graded right ideal P = (cid:18) K (cid:19) of R is graded weakly prime and P = P = 0 .But P is not graded prime right ideal of R . It should be noted that a proper graded ideal P with property that P = { } need not be graded weakly prime; see the following example: Example 2.20.
Consider the ring R = (cid:18) Q R Q (cid:19) and G = Z . Then R is G -graded by R = (cid:18) Q Q (cid:19) , R = (cid:18) R (cid:19) and R = R = 0 . Then the gradedideal P = (cid:18) R (cid:19) of G ( R ) satisfies P = { } . But P is not graded weakly primeideal of G ( R ) , since (cid:18) (cid:19) = (cid:18) (cid:19) (cid:18) Q R Q (cid:19) (cid:18) (cid:19) ⊆ P . Definition 2.21.
Let R be a graded ring. A non-empty set S ⊆ h ( R ) − { } is calleda graded weakly system if for graded ideals I and J of R with I T S = ∅ , J T S = ∅ and IJ = 0 , we have IJ T S = ∅ . Proposition 2.22.
For a proper graded ideal P of R , S = h ( R ) − P is a gradedweakly system if and only if P is a graded weakly prime ideal of R .Proof. Suppose that P is a graded weakly prime ideal of R . Let I and J be gradedideals in R such that I T S = ∅ , J T S = ∅ and IJ = 0. If IJ T S = ∅ , then IJ ⊆ P . Since P is graded weakly prime, and IJ = 0, I ⊆ P or J ⊆ P . It followsthat I T S = ∅ or J T S = ∅ , which is a contradiction. Therefore, S is a gradedweakly system in R . Conversely, suppose that IJ ⊆ P and IJ = 0, where I and J are graded ideals of R . If I * P and J * P , then I T S = ∅ and J T S = ∅ . Since S is a graded weakly system, IJ T S = ∅ , which is a contradiction. Therefore, P isa graded weakly prime ideal of R . (cid:3) Proposition 2.23.
Let S be a graded weakly system, and let P a graded ideal of R maximal with respect to the property that P is disjoint from S . Then P is a gradedweakly prime ideal.Proof. Suppose that 0 = IJ ⊆ P , where I and J are graded ideals of R . If I * P and J * P , then by the maximal property of P , we have, ( P + I ) T S = ∅ and( P + J ) T S = ∅ . Furthermore, 0 = IJ ⊆ ( P + I )( P + J ) ⊆ P . Thus, since S is agraded weakly system, ( P + I )( P + J ) T S = ∅ and it follows that ( P + I )( P + J ) * P .For this to happen, we should have IJ * P , which is a contradiction. Thus, P shouldbe a graded weakly prime ideal. (cid:3) A. S. ALSHEHRY AND R. DAWWAS
Definition 2.24.
Let R be a graded ring. For a graded ideal I of R , if there is agraded weakly prime ideal containing I , then we define GW ( I ) = { a ∈ h ( R ) : every graded weakly system containing a meets I } .If there is no graded weakly prime ideal containing I , then we put GW ( I ) = R . For a graded ideal I of R , observe that I and GW ( I ) are contained in preciselythe same graded weakly prime ideals of R . Theorem 2.25.
Let I be a graded ideal of R . Then either GW ( I ) = R or GW ( I ) equals the intersection of all graded weakly prime ideals of R containing I .Proof. Suppose that GW ( I ) = R . This means that { P : P is a graded weakly prime ideal of R and I ⊆ P } 6 = ∅ . We first prove that GW ( I ) ⊆ { P : P is a graded weakly prime ideal of R and I ⊆ P } . Let m ∈ GW ( I )and P be any graded weakly prime ideal of R containing I . Consider the gradedweakly system h ( R ) − P . This graded weakly system cannot contain m , for other-wise it meets I and hence also P . Therefore, we have m ∈ P . Conversely, assumethat m / ∈ GW ( I ). Then, by Definition 2.24, there exists a graded weakly system S containing m which is disjoint from I . By Zorn’s Lemma, there exists a gradedideal P ⊇ I which is maximal with respect to being disjoint from S . By Proposition2.23, P is a graded weakly prime ideal of R and we have m / ∈ P , as needed. (cid:3) Example 2.26.
Consider the ring R = (cid:26)(cid:18) x y (cid:19) : x, y ∈ Z , b ∈ { , } (cid:27) and G = Z . Then R is G -graded by R = (cid:18) x
00 0 (cid:19) , R = (cid:18) y (cid:19) and R = R = 0 . R has two proper graded ideals P = (cid:26)(cid:18) (cid:19) , (cid:18) (cid:19)(cid:27) and P = (cid:26)(cid:18) (cid:19) , (cid:18) (cid:19)(cid:27) . P is a graded weakly prime ideal which is not a gradedprime ideal since P P = (cid:18) (cid:19) ⊆ P but P * P . GW ( P ) = P and GW ( P ) = R . Graded Weakly Total Prime Ideals
In this section, we introduce and study the concept of graded weakly total primeideals. Recall that in [1], a proper graded ideal P of R is a graded total prime idealif xy ∈ P implies x ∈ P or y ∈ P for x, y ∈ h ( R ). Definition 3.1. (1)
A proper graded ideal P of R is graded weakly total primeif = xy ∈ P implies x ∈ P or y ∈ P for x, y ∈ h ( R ) . (2) Let R be a G -graded ring, P be a graded ideal of R and g ∈ G such that P g = R g . Then P is said to be a g -weakly total prime ideal of R if whenever x, y ∈ R g such that = xy ∈ P , then either x ∈ P or y ∈ P . The following example shows that not every graded weakly prime ideal is a gradedweakly total prime ideal.
Example 3.2.
Consider the ring of × matrices with integer entries R = M ( Z ) and G = Z . Then R is G -graded by R = (cid:18) Z Z (cid:19) , R = (cid:18) ZZ (cid:19) and R = R = 0 . Consider the graded ideal P = M (2 Z ) of R . Clearly, P is a graded RADED WEAKLY PRIME IDEALS OF NON-COMMUTATIVE RINGS 9 prime ideal and hence also graded weakly prime ideal of R . On the hand, P is notgraded weakly total prime since (cid:18) (cid:19) (cid:18) (cid:19) = (cid:18) (cid:19) ∈ P . The following three examples shows that not every graded weakly total primeideal is a graded total prime ideal.
Example 3.3.
Consider the ring R = (cid:18) Z Z Z (cid:19) and G = Z . Then R is G -gradedby R = (cid:18) Z Z (cid:19) , R = (cid:18) Z (cid:19) and R = R = 0 . Clearly, (cid:26)(cid:18) (cid:19)(cid:27) is a graded weakly total prime ideal of R which is not graded total prime since (cid:18) (cid:19) (cid:18) (cid:19) = (cid:18) (cid:19) . Example 3.4. the graded ideal P in Example 2.26 is a graded weakly total primeideal which is not graded total prime since (cid:18) (cid:19) (cid:18) (cid:19) = (cid:18) (cid:19) ∈ P . Example 3.5.
Let M be a G -graded left R -module. Let Z ( R ) be the set of allzero-divisors of R and (0 : R M ) = { a ∈ R : aM = 0 } the annihilator of M in R .Suppose that = Z ( R ) ⊆ (0 : R M ) . Let [ R, M ] = { ( a, m ) : a ∈ R and m ∈ M } bethe ring with componentwise addition and multiplication ( a, m )( b, n ) = ( ab, an ) . Infact, [ R, M ] is G -graded ring by [ R, M ] g = [ R g , M g ] for all g ∈ G , one can observethat [ R, M ] g [ R, M ] h = [ R g , M g ][ R h , M h ] = [ R g R h , R g M h ] ⊆ [ R gh , M gh ] = [ R, M ] gh for all g, h ∈ G . Now [0 , M ] is a graded ideal of [ R, M ] . In fact, it is a graded weaklytotal prime ideal, but not a graded total prime ideal. Definition 3.6. (1)
Let R be a graded ring, x, y ∈ h ( R ) and P be a gradedweakly total prime ideal of R . We say that ( x, y ) is a total homogeneoustwin-zero of P if xy = 0 , x / ∈ P and y / ∈ P . (2) Let R be a graded ring, P be a g -weakly total prime ideal of R and x, y ∈ R g .We say that ( x, y ) is a g -total twin-zero of P if xy = 0 , x / ∈ P and y / ∈ P . Note that if P is a graded weakly total prime (a g -weakly total prime) ideal of R that is not a graded total prime (not a g -total prime) ideal, then P has a totalhomogeneous twin-zero ( x, y ) for some x, y ∈ h ( R ) (a g -total twin-zero ( x, y ) forsome x, y ∈ R g ). Lemma 3.7.
Let P be a g -weakly total prime ideal of R and suppose that ( x, y ) isa g -total twin-zero of P . Then xP g = P g y = 0 .Proof. Suppose that xP g = 0. Then there exists p ∈ P g such that xp = 0. Hence x ( y + p ) = 0. Since x / ∈ P and P is g -weakly total prime, we have x + p ∈ P , andhence x ∈ P , which is a contradiction. Thus xP g = 0. Similarly, it can be easilyverified that P g y = 0. (cid:3) Lemma 3.8.
Let P be a g -weakly total prime ideal of R and suppose that ( x, y ) isa g -total twin-zero of P . If xr ∈ P for some r ∈ R g , then xr = 0 .Proof. Suppose that 0 = xr . Then 0 = xr ∈ P , and then r ∈ P since P is g -weaklytotal prime and ( x, y ) is a g -total twin-zero of P . Now, since xr ∈ xP , we have that xr = 0 by Lemma 3.7, which is a contradiction. (cid:3) Theorem 3.9.
Let R be a G -graded ring and g ∈ G . If P is g -weakly total primeideal of R but not g -total prime, then P g = 0 . Proof.
Let ( x, y ) be a g -total twin-zero of P . Suppose that pq = 0 for some p, q ∈ P g .Then by Lemma 3.7, we have 0 = ( x + p )( y + q ) ∈ P . Thus ( x + p ) ∈ P or ( y + q ) ∈ P and hence x ∈ P or y ∈ P , which is a contradiction. Therefore P g = 0. (cid:3) Corollary 3.10.
Let R be a G -graded ring and let P a graded ideal of R . If P g = 0 for some g ∈ G , then P is a g -total prime ideal of R if and only if P is a g -weaklytotal prime ideal of R . It should be noted that a proper graded ideal P with property that P = { } need not be graded weakly total prime; see the following example: Example 3.11.
Consider the ring R = (cid:18) Q R Q (cid:19) and G = Z . Then R is G -graded by R = (cid:18) Q Q (cid:19) , R = (cid:18) R (cid:19) and R = R = 0 . Then the gradedideal P = (cid:18) R (cid:19) of R satisfies P = { } . But P is not graded weakly totalprime ideal of R , since (cid:18) (cid:19) = (cid:18) (cid:19) (cid:18) (cid:19) = (cid:18) (cid:19) ∈ P . Proposition 3.12.
Let P be a g -weakly total prime ideal of R . If x ∈ R g and Y ⊆ R g such that = xY ⊆ P , then either x ∈ P or Y ⊆ P .Proof. Suppose that x / ∈ P . For every b ∈ Y such that 0 = xb ∈ P , we have b ∈ P since P is g -weakly total prime. If y ∈ Y such that 0 = xy ∈ P and y / ∈ P , then( x, y ) is a g -total twin-zero of P . Because xY ⊆ P , it follows by Lemma 3.8 that xY = 0, which is a contradiction and therefore y ∈ P and we have Y ⊆ P . (cid:3) Theorem 3.13.
For a graded ideal P of R and g ∈ G such that P g = R g , thefollowing statements are equivalent: (1) P is a g -weakly total prime ideal of R . (2) For any subset Y of R g such that Y * P , ( P : R g Y ) = { x ∈ R g : xY ⊆ P } = P S (0 : R g Y ) . (3) For any subset Y of R g such that Y * P , ( P : R g Y ) = P or ( P : R g Y ) =(0 : R g Y ) .Proof. (1) ⇒ (2): Let x ∈ ( P : R g Y ). Now xY ⊆ P . If xY = 0, then since P is g -weakly total prime it follows by Proposition 3.12 that x ∈ P . If xY = 0, then x ∈ (0 : R g Y ). So, ( P : R g Y ) ⊆ P S (0 : R g Y ). As the reverse containment holds forany graded ideal P , we have equality.(2) ⇒ (3): Since P and (0 : R g Y ) are both subgroups of R , it follows that either( P : R g Y ) = P or ( P : R g Y ) = (0 : R g Y ).(3) ⇒ (1): Let x, y ∈ R g such that 0 = xy ∈ P . If y ∈ P , then we are done.So, suppose that y ∈ R g − P . Then ( P : R g y ) = (0 : R g y ) and from (3), we have( P : R g y ) = P . Hence, x ∈ P and we are done. (cid:3) Corollary 3.14.
For a graded ideal P of R and g ∈ G such that P g = R g , thefollowing statements are equivalent: (1) P is a g -weakly total prime ideal of R . (2) For x ∈ R g − P , ( P : R g x ) = P S (0 : R g x ) . (3) For x ∈ R g − P , ( P : R g x ) = P or ( P : R g x ) = (0 : R g x ) . Recall that in [3], if T and L are two G -graded rings, then R = T × L is a G -gradedring by R g = T g × L g for all g ∈ G . RADED WEAKLY PRIME IDEALS OF NON-COMMUTATIVE RINGS 11
Theorem 3.15.
Let T and L be two G -graded rings with unities and R = T × L .If P is a graded weakly total prime ideal of R , then either P = 0 or P is a gradedtotal prime ideal of R .Proof. Assume that P = P × P where P is a graded ideal of T and P is a gradedideal of L . Suppose that P = 0. Then there is an element ( x, y ) of P such that( x, y ) = (0 , g ∈ G such that ( x g , y g ) = ( x, y ) g = (0 , P and P are graded, x g ∈ P and y g ∈ P . Now, (0 , = ( x g , y g ) = ( x g , , y g ) ∈ P and P is graded weakly total prime gives that ( x g , ∈ P or (1 , y g ) ∈ P . Supposethat ( x g , ∈ P . Then 0 × L ⊆ P , so P = P × L . We show that P is a gradedtotal prime ideal of T . Let pq ∈ P , where p, q ∈ h ( T ). Then (0 , = ( pq,
1) =( p, q, ∈ P . Now P is graded weakly total prime gives ( p, ∈ P or ( q, ∈ P .Hence, p ∈ P or q ∈ P . So, P is a graded total prime ideal of T . The case(1 , y ) ∈ P is similar. (cid:3) Recall that in [3], if R is a G -graded ring and I is a graded ideal of R , then R/I is a G -graded ring by ( R/I ) g = ( R g + I ) /I for all g ∈ G . Proposition 3.16.
Let R be a graded ring and I , P be proper graded ideals of R such that I ⊆ P . Then the following holds: (1) If P is a graded weakly total prime ideal of R , then P/I is a graded weaklytotal prime ideal of
R/I . (2) If I is a graded weakly total prime ideal of R and P/I is a graded weaklytotal prime ideal of
R/I , then P is a graded weakly total prime ideal of R .Proof. (1) Let 0 = ( x + I )( y + I ) = ( xy + I ) ∈ P/I where x, y ∈ h ( R ), so xy ∈ P .If xy = 0 ∈ I , then ( x + I )( y + I ) = 0, which is a contradiction. Hence, xy = 0 and since xy ∈ P and P is graded weakly total prime, we get x ∈ P or y ∈ P . Hence, ( x + I ) ∈ P/I or ( y + I ) ∈ P/I as required.(2) Let 0 = xy ∈ P where x, y ∈ h ( R ), so that ( x + I )( y + I ) ∈ P/I . If xy ∈ P ,then since I is graded weakly total prime, we get x ∈ I ⊆ P or y ∈ I ⊆ P .If xy / ∈ I , then 0 = ( x + I )( y + I ) ∈ P/I . Now, since
P/I is graded weaklytotal prime, we get ( x + I ) ∈ P/I or ( y + I ) ∈ P/I . Hence, x ∈ P or y ∈ P as needed. (cid:3) Corollary 3.17.
Let P and P be graded weakly total prime ideals of R that arenot graded total prime. Then P + P is a graded weakly total prime ideal of R .Proof. Since ( P + P ) /P ∼ = P / ( P T P )we get that ( P + P ) /P is graded weaklytotal prime by Proposition 3.16 (1). Now the assertion follows from Proposition 3.16(2). (cid:3) We are interested in the structure of graded rings in which every graded ideal isgraded weakly total prime. Note that by definition, a graded weakly total primeideal is a proper graded ideal of a graded ring. It is therefore not possible thatevery graded ideal of a graded ring is a graded weakly total prime ideal. However,a graded ring whose zero ideal is graded total prime is called a graded total primering. In this sense, every graded ring is a graded weakly total prime ring since thezero ideal is always graded weakly total prime. We may therefore say that everygraded ideal of a graded ring is graded weakly total prime when every proper gradedideal of the graded ring is a graded weakly total prime ideal. If R = 0, then it isevident that every graded ideal of R is graded weakly total prime. Recall that, for a ring R and x ∈ R , h x i = ( n X i =1 r i xs i + rx + xs + mx : n ∈ N , m ∈ Z , r i , s i , r, s ∈ R ) .If R has a unity, then h x i = ( n X i =1 r i xs i : n ∈ N , r i , s i ∈ R ) . Proposition 3.18.
Every graded ideal of R is graded weakly total prime if and onlyif for every a, b ∈ h ( R ) we have h ab i = h a i , h ab i = h b i or h ab i = 0 .Proof. Suppose that every graded ideal of R is a graded weakly total prime idealand let a, b ∈ h ( R ). If h ab i = R , then h a i = h b i = R . Suppose that h ab i 6 = R and h ab i 6 = 0, then h ab i is a graded weakly total prime ideal. Now, since 0 = ab ∈ h ab i ,we have a ∈ h ab i or b ∈ h ab i . Hence, h ab i = h a i or h ab i = h b i . Conversely, let I beany proper graded ideal of R and suppose that 0 = ab ∈ I for a, b ∈ h ( R ). Now wehave h a i = h ab i ⊆ I or h b i = h ab i ⊆ I . Hence, a ∈ I or b ∈ I and we are done. (cid:3) Corollary 3.19.
Let R be a graded ring in which every graded ideal is graded weaklytotal prime. Then for every x ∈ h ( R ) , h x i = h x i or h x i = 0 . The next example shows that the converse of Corollary 3.19 is not true in general.
Example 3.20.
Let K be a field and R = K L K L K . Then for every element x ∈ R , we have h x i = h x i . On the other hand, if we consider the trivial graduationfor R by a group G , then P = K L L will be a graded ideal of R which is notgraded weakly total prime. Acknowledgement
This research was funded by the Deanship of Scientific Research at PrincessNourah bint Abdulrahman University through the Fast-track Research Funding Pro-gram.
References [1] R. Abu-Dawwas, M. Bataineh and M. Al-Muanger, Graded prime submodules over non-commutative rings, Vietnam Journal of Mathematics, 46 (3) (2018), 681-692.[2] S. E. Atani, On graded weakly prime ideals, Turkish Journal of Mathematics, 30 (2006), 351-358.[3] C. Nastasescu and F. V. Oystaeyen, Methods of graded rings, Lecture Notes in Mathematics,1836, Springer-Verlag, Berlin, (2004).[4] M. Refai, M. Hailat and S. Obiedat, Graded radicals and graded prime spectra, Far EastJournal of Mathematical Sciences, (2000), 59-73.[5] R. N. Uregen, U. Tekir, K. P. Shum and S. Koc, On Graded 2-Absorbing Quasi Primary Ideals,Southeast Asian Bulletin of Mathematics, 43 (4) (2019), 601-613.
Azzh Saad Alshehry, Department of Mathematical Sciences, Faculty of Sciences, Princess NourahBint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia.Email address : [email protected] Rashid Abu-Dawwas, Department of Mathematics, Yarmouk University, Irbid, Jordan.Email address ::