Abstract

Let be a commutative ring with nonzero identity and, be a multiplicatively closed subset. An ideal of is called an -quasi-primary ideal if and there exists an (fixed) and whenever for then either or In this paper, we construct a topology on the set of all -quasi-primary ideals of which is a generalization of the -prime spectrum of . Also, we investigate the relations between algebraic properties of and topological properties of like compactness, connectedness and irreducibility.

1. Introduction

All rings considered below are commutative with nonzero identity. The concept of prime ideals and their generalizations play an important role in Commutative Algebra and Algebraic Geometry since they are used to characterize some classes of rings and they have some applications to other areas of mathematics such as General Topology and Graph Theory (see for instance, [19]). A multiplicative closed subset (briefly, m.c.s) of a commutative ring is a subset of such that , and . Let be a commutative ring and be a m.c.s of . Recall [10, 11] that an ideal of is said to be -prime if and there exists a fixed such that whenever for some , then either or . The set of -prime ideals of is denoted by . The concept of -prime ideals is a generalization of prime ideals. Recall from [12] that an ideal is said to be quasi-primary if its radical is prime. The notion of quasi-primary ideals generalizes prime and primary ideals. By analogy, our aim in this paper is to generalize naturally the notion of -prime ideals. In fact, if is a commutative ring and is a m.c.s of , we say that an ideal of is -quasi primary if and there exists a fixed such that whenever for some , then either or . We let denote the set of all -quasi primary ideals of . Then we equip with a natural topology whose basis opens are of the form , where . The space is called the -quasi-primary spectrum of . This topology is a natural generalization of the -Zariski topology introduced in [13]. More precisely, the -prime spectrum is a dense subspace of , see Corollary 2. The basic properties of the -quasi-primary spectrum are studied. It is shown that the -quasi-primary spectrum satisfies all the conditions of being a spectral space except the uniqueness of generic point, see Proposition 6, Lemma 2 and Theorem 4. Any unexplained terminology is standard as in [14, 15].

2. S-Quasi Primary Ideals

Throughout this paper, will denote a m.c.s of a commutative ring . Recall from [13] that the set of all -prime ideals of is denoted by . In [12], L. Fuchs introduced the concept of quasi-primary ideals. More precisely, an ideal of is said to be quasi-primary if is a prime ideal. By analogy, we introduce the following definition.

Definition 1. Let be a commutative ring and a m.c.s of . An ideal of a ring is said to be -quasi primary if . In other words, is -quasi-primary, if and there exists an (fixed) such that whenever for some , then either or .
We let denote the set of all -quasi primary ideals of . It is worth noticing that is always nonempty because and so there exists a prime ideal of disjoint from according to ([16], Theorem 3.44). Therefore, . The following result follows immediately from ([17], Lemma 2.3).

Proposition 1. Let be a commutative ring and a m.c.s of . Then.

Let be an ideal of and be a nonempty subset of . The residual of by is defined as and also for the singleton , we prefer instead of . Recall, from ([11], Lemma 2.16 and Theorem 2.18) (see also ([13], Proposition 2)), that if is an -prime ideal of for a m.c.s of , then there exists an element such that for all . Moreover, is a prime ideal of . Using this fact we get immediately the following result which will be frequently used in the sequel.

Proposition 2. Letbe a commutative ring anda m.c.s ofIf , then there exists :such that(1) for all.(2) is a prime ideal of.

3. Quasi Zariski Topology

Let be a commutative ring and a m.c.s of . In [13], the authors have constructed a topology on called the -Zariski topology. The closed subsets of are the -varietieswhere is any subset of . Thus, any open subset has the form for any subset . The collection , where , constitutes a basis for the -Zariski topology. Our aim here is to built a natural topology on the set of all -quasi primary ideals. To this end, we consider the following mapping:and we equip with the topology , which is, in fact, the initial topology with respect to the mapping . We call this topology the -quasi Zariski topology. For any subset of , we let . It follows that any closed subset for the -quasi Zariski topology has the form , where is any subset of . If , then we will let , , and denote respectively, , , and .

We start our investigation with the following result.

Theorem 1. Let be a commutative ring and a m.c.s of :Then the following hold true(1) for any subset of , where denotes the ideal generated by the set.(2) and .(3) for any family of ideals of .(4) for any two ideals and of .(5)If , then for any two ideals andof.

Proof. Statements (1)-(4) follow from ([13], Theorem 1) by taking the inverse images of the ’s by the mapping . Statement (5) comes directly from ([13], Remark 2).
We derive from ([13], Theorem 2) the following result.

Proposition 3. Let be a commutative ring and a m.c.s of . A basis for is the collection of all , where .

Proposition 4. Let be a commutative ring and a m.c.s of . If is an ideal of , then .

Proof. Let . Then there exists such that . Thus, . Hence, . Conversely, let . Then there exists such that . Let . There exists an integer such that . Hence, . As , then is a prime ideal of . Therefore, . It follows that . Hence, . This completes the proof.

Corollary 1. Let be a commutative ring and a m.c.s of . Then for any .

Proof. Follows from Proposition 4 since and .

Corollary 2. is a dense subspace of .

The following proposition determines the closure of any subset of .

Proposition 5. Let be a commutative ring and a m.c.s of . Then for any , we have:

Proof. Let . As, then . Thus, . It follows that . Hence, . Conversely, suppose for some subset of . Thus, for any , there exists an element such that . This yields that . It follows from Proposition 2 that . This shows that . Now, Theorem 1 ensures that . Since is the smallest closed set containing , then we get immediately . This completes the proof.
In Proposition 6, we will characterize the closure of the singleton for all . But first we need the following useful lemma.

Lemma 1. Let be a commutative ring and a m.c.s of . Then the following hold true:(1) for any ideals , of and for any .(2) for any ideal of .(3).

Proof. (1) Let and . Then there exists an integer such that . As , then . Hence, for some integer . This implies that . Hence .
(2) Follows by combining Theorem 1 (5) and assertion (1).
(3) This is clear since for any ideal of

Corollary 3. Let be a closed subset of . If , then .
The next result shows among other facts that generic points in are not unique.

Proposition 6. Let be a commutative ring, a m.c.s of and . Then the following hold true:(1).(2)Suppose that , then

Proof. (1) It follows from Proposition 5 that . Now we will show that . Since , then by Theorem 1. Conversely, let . Then there exists such that . This implies , namely . The equality follows from Lemma 1 Finally, by the fact that .
(2) Since , then . The result follows from assertion (1) and the proof of ([13], Proposition 4 (ii)).
Notice that if is a (unital) ring homomorphism and is a m.c.s of , then is a m.c.s of if and only if .

Proposition 7. Let be a (unital) ring homomorphism and a m.c.s of with . If , then .

Proof. As , then . Thus, by ([13], p. 1216). As is a ring homomorphism, then it is clear that for any ideal of the ring . It follows that and so . This completes the proof.

Theorem 2. Let be a (unital) ring homomorphism and a m.c.s of with . The map , defined by , is continuous.

Proof. Let be a subset of . We have:This shows that is continuous.

4. Some Results on the -Radical

Recall from ([13], Definition 1) that if is a commutative ring, is a m.c.s of and is an ideal of , then the -radical of is defined as follows:

The main purpose of this section is to provide some new results concerning the -radical. We start with the following result.

Proposition 8. Let be a commutative ring and a m.c.s of . Let such that . Then

Proof. Let and let . Then . Thus, there exists an integer such that . Hence, . But as is a prime ideal of , then we get . It follows that . This shows that. Conversely, let and let . Since , then . Hence, and so . This yields that. Therefore,. This completes the proof.
Let be a commutative ring, a m.c.s of and an ideal of . It was proved in ([13], Proposition 5) that. The next result shows in fact that .

Corollary 4. Let be a commutative ring and a m.c.s of . If is an ideal of , then

Proof. According to ([13], Proposition 5), . As (Proposition 4) and by Corollary 3, we derive from Proposition 8that.

Proposition 9. Let be a commutative ring and a m.c.s of . If is an ideal of , then .

Proof. As , then by virtue of Theorem 1. Conversely, let , then for some . Let . Then for some and . Hence, . But as is a prime ideal, then . So , which shows that . The proof is complete.

Corollary 5. Let be a commutative ring and a m.c.s of . If is an ideal of , then .

Proof. It follows from Proposition 1 that . Combining Proposition 5 and Proposition 8 we get:Hence, by Proposition 9.

5. Irreducibility of

Recall that a topological space is called irreducible if is nonempty and cannot be expressed as the union of two proper closed subsets, or equivalently, any two nonempty open subsets of intersect (cf. [15, 18]).

Proposition 10. Let be a commutative ring and a m.c.s of . If , then is an irreducible set in .

Proof. Let . Since , then is irreducible by virtue of ([13], Proposition 8). Hence is irreducible according to Corollary 5. But by Proposition 6. Therefore, is also irreducible.

Proposition 11. Let be a commutative ring and a m.c.s of . If is an ideal of such that is irreducible, then .

Proof. As is irreducible, then so is by Corollary 5. Thus, ([13], Proposition 9) guarantees that .
Recall from ([13], Remark 3) that is called the -nilradical of and denoted by .

Lemma 2. Let be a commutative ring and a m.c.s of . For any , we have:(1).(2).

Proof. (1) Follows from ([13], Proposition 6 (i)) and from the fact that for any .
(2) If , then by Corollary 1. Thus, by virtue of ([13], Proposition 6 (ii)). Conversely, if , then ([13], Proposition 6 (ii)) ensures that . Hence, .
For any ring , we let denote the nilradical of ; that is, the set of all nilpotent elements.

Theorem 3. Let be a commutative ring and a m.c.s of . Consider the following conditions:(1) is an irreducible space;(2);(3);(4).Then (2) (1). If moreover, , then (4) (3).

Proof. (2) (3) This clear, since .
(3) (4) Let such that . It follows from ([17], Lemma 2.1) that there exists such that . As , there exists an (fixed) such that either or . Using again ([17], Lemma 2.1), we infer that either or . Therefore, .
(4) (1) Let . By assumption, is a prime ideal of . Thus, . By Propositions 9 and 10, we deduce that is an irreducible space.
(1) (4) Assume that is an irreducible space. Set . Then is irreducible. By Proposition 11, .
To prove the “moreover” statement, it is enough to show that if , then . To this end, let . By using, ([17], Lemma 2.1), there exists an element such that . So there exists an integer such that . As is a regular element in , we get . Hence, . This proves that . As the reverse inclusion always holds, we get the equality.
Recall from [11] that a ring is said to be an -integral domain if there exists a fixed and whenever for some , then either or . Note that every integral domain is an -integral domain. Also a ring is an -integral domain if and only if the zero ideal is an -prime ideal.

Corollary 6. Let be a commutative ring and a m.c.s of . If is an -integral domain, then is an irreducible space.

Proof. As is an -integral domain, then by using [13, Corollary 1], is irreducible. But, Corollary 2 guarantees that is a dense subspace of . Thus, is an irreducible space.

6. Compactness and Connectedness of

Recall from [15] that a topological space is called compact if any open cover of has a finite subcover.

Lemma 3. Let be a commutative ring and a m.c.s of . For any , we have:

Proof. Note that . Thus, the result follows immediately from ([13], Proposition 6 (iii)).

Theorem 4. Let be a commutative ring and a m.c.s of . Then is compact for any . In particular, is compact.

Proof. Assume that , where for any . Then, by using Theorem 1, we obtain, where is the ideal of generated by the ’s. It follows from Proposition 4 that . Therefore, ([17], Lemma 3.1) ensures that . Hence, . Thus, there exist and such that . Therefore, there exists a finite subset of such that . This shows that . Thus, by virtue of Theorem 1. Hence, (see Proposition 9); or equivalently, . It follows that as desired.
As a consequence, we derive the following corollary.

Corollary 7. Let be a commutative ring and a m.c.s of . Assume that . Then is compact.

Proof. Using Theorem 4, it is enough to show that if is an open subset of containing , then . To this end, let . As , then it follows from ([16], Theorem 3.44) that there exists such that and . Since , then there exists such that . Thus, for any , . In particular, for any , . This proves that . Therefore, we have proved that . As the reverse inclusion is obvious, we get . The proof is complete.
We close the paper with the following result. But first recall [15] that a topological space is said to be connected if it cannot be written as a union of two disjoint proper open sets.

Theorem 5. Let be a m.c.s of . Then the following statements are equivalent:(1) is a connected space.(2)For any idempotent in , either or for some

Proof. (1) (2) It is enough to show that is connected and then apply ([13], Theorem 8). To this end, assume the contrary. Write where and are disjoint proper open subsets of . Then, we get . This is a partition of into two proper open sets according to Corollary 1, which is a contradiction.
(2) (1) The hypotheses of assertion (2) imply that is connected. As , then clearly is connected. This completes the proof.

7. Conclusion

In this work, we have generalized the concept of quasi-primary ideals already studied by Fuchs in [12] to that of -quasi-primary ideals, where is a m.c.s of a commutative unital ring . We have equipped the set of -quasi-primary ideals of with a topology called the -quasi-Zariski topology, which is a natural generalization of the -Zariski topology, already investigated by Yildiz et al. in [13]. Several satisfactory results are obtained. We have proved that this topology satisfies all the conditions of being a spectral space except the uniqueness of generic point. Concerning the future plans for our work, we will study topological spaces which are -spectral or -Zariski spectral, in the sense that they are homeomorphic to the -spectrum or the -quasi-spectrum of a commutative ring . Another direction for research, is to provide conditions on under which is a normal, totally disconnected, extremely disconnected topological space.

Data Availability

All data required for this paper are included within this paper.

Conflicts of Interest

The authors declare no conflicts of interest.

Acknowledgments

This work was supported through the Annual Funding track by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia (Project No. AN000161, GRANT935). The authors would like to thank the referees for carefully reading our manuscript and for giving such constructive comments which substantially helped improving the quality of the paper.