Properties of -Primal Graded Ideals

Abstract

Let be a commutative graded ring with unity . A proper graded ideal of is a graded ideal of such that . Let be any function, where denotes the set of all proper graded ideals of . A homogeneous element is -prime to if where is a homogeneous element in ; then . An element is -prime to if at least one component of is -prime to . Therefore, is not -prime to if each component of is not -prime to . We denote by the set of all elements in that are not -prime to . We define to be -primal if the set (if ) or (if ) forms a graded ideal of . In the work by Jaber, 2016, the author studied the generalization of primal superideals over a commutative super-ring with unity. In this paper we generalize the work by Jaber, 2016, to the graded case and we study more properties about this generalization.

1. Introduction

In [1] the author studied the generalization of primal superideals over a commutative super-ring with unity. In this paper we generalize this work to the graded case and we study more properties about this generalization. For example in Section 4 we study the properties of --primal graded ideals and in Section 5 we study the properties of -primal graded ideals of .

Let be an abelian group and let be any commutative ring with unity; then is called a -graded ring (for short graded ring), if , such that if , then . Let ; then is the set of homogeneous elements in and , where is the identity element in . By a proper graded ideal of we mean a graded ideal of such that .

We define a proper graded ideal of to be prime if implies that or , where . Let be a proper graded ideal of , an element is called prime to if , where ; then . If is a proper graded ideal of and is the set of homogeneous elements of that are not prime to , then we define to be primal if the set forms a graded ideal of . In this case we say that is a -primal graded ideal of . Moreover, if is a -primal graded ideal of , then it is easy to check that is a prime graded ideal of .

Throughout, will be a commutative graded ring with unity. We will denote the set of all proper graded ideals of by . If and are in , then the set is a graded ideal of which is denoted by . Let be any function and let ; we say that is a -prime if whenever , with , then or . Since , there is no loss of generality to assume that for every proper graded ideal of .

Let be any function. In this paper we always assume that, for any , if .

Given two functions , we define if for each .

Let be any function; then an element is -prime to , if whenever , where , then . That is, is -prime to if An element is -prime to if at least one component of is -prime to . Therefore, is not -prime to if each component of is not -prime to . Denote by the set of all elements in that are not -prime to . Note that if is not -prime to , then .

We define to be -primal if the set (if ) or (if ) forms a graded ideal of . In this case we say that is a --primal graded ideal of , and is the adjoint graded ideal of .

In the next example we give some famous functions and their corresponding -primal graded ideals.

Example 1.   primal graded ideal  weakly primal graded ideal  almost primal graded ideal -almost primal graded ideal -primal graded idealObserve that .

In this paper we study various properties of -primal graded ideals. Some of these properties for the nongraded case have been studied by Atani and Darani in [2, 3].

2. -Primal Graded Ideals

The next example shows that need not to be a graded ideal of (see this example also in [4]).

Example 2. Let , where , be a commutative -graded ring and assume that . Let . Then is a graded ideal of . Since with , then we get that and are not -prime to , and hence and are in . Easy computations imply that is -prime to . Thus, which implies that is not a graded ideal of .

Next we give two examples of -primal -graded ideals of a given -graded ring (where ).

Example 3. Let , , and . Then , where , , , and for , is a -graded ring. Let be a graded ideal in , and let . Then is a graded ideal of ; hence is -primal graded ideal of .

Example 4. Let and . Then is -graded ring, where and for . Let . Then, for , by easy computations where . Hence is -primal graded ideal of .

Lemma 5. Let be a proper graded ideal of , and let be any function. Then .

Proof. Let . Since is proper, . Then implies that is not -prime to . Thus .

We recall from Lemma 5 that if each component of is in . Thus, by Lemma 5, each element can be written in the form , where and . Therefore, .

Theorem 6. Let be a proper graded ideal of , and let be any function. If the set is an ideal of , then it is -prime graded ideal of .

Proof. Clearly is a graded ideal of . Now, let with . Then is not -prime to . So there exists with . If , then is -prime to . So implies that . Hence , since if , then , a contradiction. Thus, , since .

Corollary 7. Let be a proper graded ideal of , and let be any function. If the set is an ideal of , thenis -prime graded ideal of .

Proof. let with . Then , since . Therefore, or , where and .
If , then, by Theorem 6, or . So, is -prime graded ideal of .
If , where and , then there exists with . So, , since . Therefore, and by Theorem 6, or .

Definition 8. Let be a proper graded ideal of , and let be any function. Then is called --primal graded ideal of , where , if the set forms an ideal of . By the above corollary, is always -prime graded ideal of . In this case is called the adjoint of .

Proposition 9. Let , be proper graded ideals of . Then the following statements are equivalent.(1) is -primal graded ideal of with the adjoint graded ideal .(2)For with we have . If , then .

Proof. If , then . So there exists with . Thus and . Since it is easy to see that , we have that .
Now let , where . Then and hence is -prime to . Let . If , then . If , then . Hence From part we have . Thus is -primal graded ideal of .

We say that is -primal graded ideal of if itself is the adjoint of . The next result shows that every -prime graded ideal of is -primal.

Theorem 10. Every -prime graded ideal of is -primal.

Proof. Let be -prime graded ideal of ; we show that is --primal graded ideal of . Thus we must prove thatIf , then it is easy to check that ; hence is --primal graded ideal of . Therefore, we may assume that . We show that . Let . Then with , so . On the other hand let . If , then for all , so is -prime to and hence . If , then , so for any with we have that , since is -prime. Thus is -prime to and hence . Therefore, which implies that is --primal graded ideal of .

Now we give an example of --primal graded ideal of such that itself is not -prime.

Example 11. Let and let where . Then is a commutative -graded ring with unity. If , then is not a -prime graded ideal of , since , but . Let ; we show that is --primal graded ideal of . It is enough to show that . Let , if ; then . If is an odd number, then , but , and if is an even number with ; hence . If , then . On the other hand, if , then is an odd number in . If for some , then divides and so divides since ; hence . Thus is --primal graded ideal of .

Let be any function. We assume that, for any , if . Now we prove one of the main results in this section.

Theorem 12. Suppose that , where and are maps from into , and let be a --primal graded ideal of , with for all , where is the identity element in . If is a prime graded ideal of , then is --primal.

Proof. Since is --primal graded ideal of , thenTo show that is --primal graded ideal of we must prove thatIf , then and hence we have that which implies that is --primal graded ideal of . Now we may assume that . Let . Then there exists with , so which implies thatNow, let . If , then . So with . Hence . Therefore,From (7) and (8) we have thatSince , by (9)Let . Then there exists with . If , then . So we may assume that ; hence . First suppose that , say with . Then with . Hence . Therefore, we may assume that .
Now suppose that . Then there exists with . Since , then with . So but ; therefore, and hence , since is a prime graded ideal. So we may assume that . Since there exists and with . Thus, , so with which implies that . Hence , and so , since is a prime graded ideal of . Therefore, , soand hence is --primal graded ideal of .

3. Conditions on -Primal Graded Ideals

In this section we introduce some conditions under which -primal graded ideals are primal.

Let be any function. We have to recall that if is --primal graded ideal of , thenis -prime graded ideal of .

Definition 13. Let be a homogeneous element in . Then if for some .

In the next theorem we provide some conditions under which -primal graded ideal is primal.

Theorem 14. Let be a commutative graded ring with unity and let be any function. Suppose that is --primal graded ideal of with for each . If is a prime graded ideal of , then is -primal.

Proof. Assume that is a homogeneous element in . Then or for some or where and for some . If the first two cases hold, then is not prime to , since it is not -prime to . In the last case, let be a homogeneous element in such that with . Then , because implies that , since which is a contradiction. Thus is not -prime to and hence is not prime to . Now assume that is not prime to , so for some homogeneous element . If , then is not -prime to , so . Thus assume that . Suppose that . First suppose that . Then, there exists such that . So , where is a homogeneous element in , implies that is not -prime to ; that is, . Therefore, we may assume that . Let . If , then for some . In this case with ; that is, , and hence , since . So we may assume that . Since , there are and with . Now, , where is a homogeneous element in , implies that is a homogeneous element in . On the other hand , so that . We have already shown that is exactly the set of all elements of that are not prime to . Hence is -primal.

Let and be commutative -graded rings, where is an abelian group. It is easy to prove that the prime graded ideals of have the form or where is a prime graded ideal of and is a prime graded ideal of . Also we have the following proposition about primal graded ideals of . We leave the easy proof for this proposition to the reader. For the trivial case (i.e., for all ) the next proposition is proved in [5, Lemma ].

Proposition 15. Let and be commutative -graded rings. If is -primal graded ideal of and is -primal graded ideal of , then (resp., ) is - (resp., -) primal graded ideal of .

Next, we generalize [6, Theorem ] to the graded case. Then we use this generalization to prove Theorem 18.

Theorem 16. Let and be commutative -graded rings with unities and let be functions. Let . Then -primes of have exactly one of the following three types:(1) where is a proper graded ideal of with (2) where is -prime of which must be prime if (3) where is -prime of which must be prime if

Proof. We first note that a graded ideal of having one of these three types is -prime. Case is clear since . If is a prime graded ideal of , certainly is prime and hence -prime. So suppose that is -prime and . Let , be homogeneous elements in and let , be homogeneous elements in with . Then or , so or . So, is -prime. The proof for case is similar.
Next, suppose that is -prime. Let , be homogeneous elements in such that . Then , so or ; that is, or . So is -prime. Likewise, is -prime. Suppose that , say, . Let be a homogeneous element in such that and let be a homogeneous element in . Then . So or . Hence or . Suppose that . So is -prime where is -prime. It remains to show that if , then is actually prime graded ideal of . Let , be homogeneous elements in such that . Now is a homogeneous element in not in . Then , so or ; that is, or .

Let , be commutative -graded rings with unities and let . Let be a function. In the next theorem we also provide some conditions under which -primal graded ideal of is primal, but first we start with the following remark.

Remark 17. Let be a proper graded ideal of a commutative graded ring and let be a function. If a homogeneous element is not -prime to , then there is a homogeneous element in such that so is not prime to .

Theorem 18. Let , be commutative -graded rings with unities and let . Let be functions with for . Let . Assume that is a graded ideal of with . If is a --primal graded ideal of , then either or is -primal.

Proof. Suppose . By Corollary 7, is -prime graded ideal of . Therefore, by Theorem 16, has one of the following three cases.
Case 1. , where is a proper graded ideal of with for . In this case , a contradiction.
Case 2. where is -prime graded ideal of . Since , by Theorem 16, is a prime graded ideal of and so is a prime graded ideal of . We show that . Since , there exists a homogeneous element in . So . If , then is not -prime to ; hence , so , a contradiction. Thus ; that is, and . Now we prove that is -primal graded ideal of . Let be a homogeneous element in . Then . If , then so is not prime to . Therefore, we may assume that . In this case there exists a homogeneous element such that so with , since , implies that is not -prime to ; hence by Remark 17, is not prime to . Conversely, let be a homogeneous element in such that is not prime to . Then there exists a homogeneous element in with . Since , with . Hence is not -prime to which implies that and so . We have already shown that the set of homogeneous elements in consists exactly of the homogeneous elements of that are not prime to . Hence is -primal graded ideal of so by Proposition 15, is -primal graded ideal of .
Case 3. , where is -primal graded ideal of . The proof of Case3 is similar to that of Case2.

4. --Primal Graded Ideals

Let be a commutative graded ring with unity and let be a proper graded ideal of . Let be any function. As a generalization of [6], we define by for every graded ideal with (and if ).

We leave the trivial proof of the next lemma to the reader.

Lemma 19. Let be a commutative graded ring with unity and let be a proper graded ideal of . Let be any function. If is -prime graded ideal of containing , then is -prime graded ideal of .

Lemma 20. Let be a commutative graded ring with unity and let be a proper graded ideal of , and let be any function. Let be a graded ideal of containing . If is -prime graded ideal of with , then is -prime graded ideal of .

Proof. Let , be homogeneous elements in with . Then and . Thus, , so which implies that ; that is, or so or . Therefore, is -prime graded ideal of .

In the next result and under the condition that we prove that is -primal graded ideal of if and only if is -primal graded ideal of .

Theorem 21. Let be a commutative graded ring with unity and let be any function. Let be a proper graded ideal of , and let be a graded ideal of with . Then is --primal graded ideal of if and only if is --primal graded ideal of .

Proof. Suppose that is --primal graded ideal of with . Then, by Corollary 7, is -prime graded ideal of containing , so, by Lemma 19, is -prime graded ideal of . We show that is --primal graded ideal of . That is, we must prove thatLet ; then is not -prime to . So there exists with . If , then . So we may assume that . Therefore, and because we get that .
Now, assume that is a homogeneous element in such that . Then there exists a homogeneous element in such that , so with . Thus, is not -prime to which implies that , and hence . Therefore,and so is --primal graded ideal of .
Conversely, suppose that is --primal graded ideal of with the adjoint graded ideal . We show that is --primal graded ideal of . Now, by Corollary 7, is -prime graded ideal of with , so, by Lemma 20, is -prime graded ideal of . To finish the proof we need to show thatClearly, . Let ; then there exists a homogeneous element with . Since we get that and . So, and hence .
Now, let . Suppose that . Then . If , then we are done. Assume that . Then and, so, is not -prime to ; hence . Therefore, we may assume that , so there exists with , and so ; that is, . Therefore, with ; that is, .

By Theorem 21, we get the following result.

Corollary 22. Let be a commutative graded ring with unity, and let be any function. let be a graded ideal of . Then there is one-to-one correspondence between --primal graded ideals of containing with and --primal graded ideals of .

5. -Primal Graded Ideals of

Let be a commutative -graded ring (for short graded ring) with unity, let be a multiplicatively closed subset of , and denote by the ring of fractions . We define a grading on by settingIt is easy to see that is -graded ring (for short graded ring). Also, for -graded ideal of , is -graded ideal of .

Consider the canonical homomorphism which is defined by for all . Then is a homogenous graded homomorphism of degree 0.

Now let be any function, we define by for every . Note that , since, for , we have that implies .

Example 23. Let with . Let , . Then is a multiplicatively closed subset of . If , then one can easily check that is --primal graded ideal of . Moreover, ; hence , since , where is the canonical homomorphism. Therefore, is --primal graded ideal in .

We start by proving the following properties about -prime graded ideals of , where is the canonical homomorphism and is a multiplicatively closed subset of .

Lemma 24. Let be any function, and let be -prime graded ideal of with ; then is -prime graded ideal of .

Proof. Let , be homogeneous elements in with ; then, for some , , so or , and thus or ; hence is -prime graded ideal of .

Theorem 25. Let be any function, and let be -prime graded ideal of with . If , then .

Proof. It is easy to see that .
Conversely, let be a homogeneous element in , then for some , . If , then and so . Therefore, we may assume that , so is a homogeneous element in . Thus,and since , we have that .

Lemma 26. Let be any function, and let be --primal graded ideal of with . If , then . Moreover, if , then .

Proof. Let , so for some and . In this case for some . If , then a contradiction. So we have that . If , then is not -prime to ; so which contradicts the hypothesis. Therefore, .
For the last part, it is clear that . Now let be a homogeneous element in . Then for some . If and , then is not -prime to , so a contradiction. So must be in . If , then , and so . Therefore, , since . Hence .

Lemma 27. Let be any function, and let be --primal graded ideal of with . Then .

Proof. Let be a homogeneous element in such that ; then and, by Lemma 26, . If , then implies that a contradiction. Therefore, .

Let be a commutative graded ring with unity and be -graded module. An element is called a zero-divisor on if for some . We denote by the set all zero-divisors of on .

Corollary 28. Let be any function, and let be --primal graded ideal of with , . If , then .

Proof. By Lemma 26, if , then . Let be a homogeneous element in ; then , where . If , then . Therefore we may assume that . If , then Therefore we may assume that . So, is a homogeneous element in . So for some and . So there exists such that . If , then a contradiction. Therefore, . So . If , then there exists with , so a contradiction. Thus and . So, . Hence .

We recall that if is a graded ideal in , then ; therefore, we may assume that .

Under the condition that for all proper graded ideals of , we have the following propositions.

Proposition 29. Let be a multiplicatively closed subset of with , let be any function, and let be --primal graded ideal of with , . Then is --primal graded ideal of .

Proof. By Lemma 24, is -prime graded ideal of .
To show that is --primal graded ideal of , we must prove thatClearly, ; let be a homogenous element in . Then there exists with , so and, by Lemma 26, . So, , and . Thus and hence .
Conversely, let such that . Then . If , then , , so is not -prime to ; thus . Therefore, we may assume that ; that is, for every . So, . Therefore, . Thus, . Since, by Corollary 28, we have that .

Proposition 30. Let be any function, and let be --primal graded ideal of . Then is -prime graded ideal of and is -primal graded ideal of with the adjoint graded ideal with , . Moreover, .

Proof. To show that is -prime graded ideal of , it is enough to prove that is -primal graded ideal of with the adjoint graded ideal . Then, by using Corollary 7, will be -prime graded ideal of .
Now, to prove that is -primal graded ideal of we must show thatBut . Let be a homogenous element in with ; then , since and, by Corollary 28, . Thus, and hence .
Conversely, let be a homogeneous element in . Then in . We may assume that , since , . If , then and since , , but , so . If , then and so . Let be a homogeneous element in , with ; then , since and , for if , then , a contradiction. Thus we have that and , since . Therefore, and so is -primal graded ideal of with the adjoint graded ideal .
Finally, we show that . Clearly, . Conversely, let be a homogeneous element in . Then for some . Thus, , and hence . Therefore, .

Under the condition that for all proper graded ideals of and by using Propositions 29 and 30 we have the following result.

Corollary 31. Let be a commutative graded ring with unity. Let be a multiplicatively closed subset of , and let be any function. Then there is one-to-one correspondence between --primal graded ideals of and --primal graded ideals of , where is -prime graded ideal of with , .

I would like to remark that “-theory” introduced here has nothing in common with “-theories” considered previously by Badawi and Lucas in [7].

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.