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Journal of Mathematics
Volume 2017, Article ID 3817479, 8 pages
https://doi.org/10.1155/2017/3817479
Research Article

Properties of -Primal Graded Ideals

Department of Mathematics, The Hashemite University, Zarqa 13115, Jordan

Correspondence should be addressed to Ameer Jaber; oj.ude.uh@jreema

Received 28 February 2017; Revised 28 April 2017; Accepted 11 May 2017; Published 4 June 2017

Academic Editor: Naihuan Jing

Copyright © 2017 Ameer Jaber. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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,