{'fraction_non_alphanumeric': 0.06447346668870887, 'fraction_numerical': 0.008927095387667384, 'mean_word_length': 3.3789359391965257, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 8, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Quasi-Primary Spectrum of a Commutative Ring and a Sheaf of Rings

In this work, the set of quasi-primary ideals of a commutative ring with identity is equipped with a topology and is called quasi-primary spectrum. Some topological properties of this space are examined. Further, a sheaf of rings on the quasi-primary spectrum is constructed and it is shown that this sheaf is the direct image sheaf with respect to the inclusion map from the prime spectrum of a ring to the quasi-primary spectrum of the same ring.


Introduction
The set of all prime ideals of a commutative ring R, called prime spectrum of R, denoted by Spec(R), is a well-known concept in commutative algebra. This set is equipped with the famous Zariski topology, where closed sets are defined as V (I) = {P ∈ Spec(R) : I ⊆ P } for any ideal I of R. Topological properties of Spec(R) are widely examined throughout the years and can be found in many of the standard commutative algebra and algebraic geometry references. Besides, there is a famous sheaf construction, named the structure sheaf, on Spec(R) which is a very useful tool to connect algebraic geometry and commutative algebra. For details of the structure sheaf the reader may consult [6], [1] and [2].
In [5], the authors generalized the Zariski topology on Spec(R) to the set of primary ideals of a commutative ring R, denoted by P rim(R), and they called it primary spectrum of R. They defined the closed sets as V rad (I) = {Q ∈ P rim(R) : I ⊆ Q} for any ideal I of R where √ Q denotes the radical of Q. They showed that these closed sets satisfy axioms of a topology on P rim(R). They investigated some topological properties of this space and compared them with the well-known properties of Spec(R). We note that, since any prime ideal is primary and equal to its radical, the space Spec(R) is in fact a subspace of P rim(R). When [5] is examined in detail, it can be realized that the given topological construction depends only the fact that the radical of a primary ideal is prime. So, this topology is in fact valid on a much larger set, the set of ideals whose radicals are prime. These type of ideals are first introduced by L. Fuchs in [3]. He named them as quasi-primary ideals. We aim to investigate the set of quasi-primary ideals of a commutative ring R equipped with a topology similar to the one defined in [5] and to construct a sheaf of rings on this topological space.
In Section 2, after giving some general topological properties of quasi-primary spectrum, we examine irreducibility and irreducible components of this space. Besides we investigate disconnectedness of the space and finally show that the dimension of the quasi-primary spectrum of a Noetherian local ring is finite. In Section 3, we construct a sheaf of rings on the quasi-primary spectrum of and prove that this sheaf is actually the direct image sheaf under the inclusion map from the prime spectrum to the quasi-primary spectrum. Further, we conclude that this sheaf is in fact a scheme.

The Quasi-Primary Spectrum of a Ring
Throughout this paper all rings are commutative with identity. In this section, we define a topology on the set of all quasi-primary ideals of a ring and examine some properties of this topological space.
First, we give some (known) properties of quasi-primary ideals we need in the rest of the paper. Let R be a ring. Following [3], an ideal I of R is called quasi-primary if the radical of I, denoted by √ I is prime. In this section  Observe that, for any subset In [5], the authors defined a topology on the set P rim(R) of primary ideals of a commutative ring using the sets V rad (I) = {Q ∈ P rim(R) : I ⊆ √ Q} as the closed sets. In this construction they only used the property that a primary ideal has a prime radical. So we realized that the topology axioms for closed sets are in fact satisfied by the sets where I any ideal of R. Thus, QP rim(R) is a topological space with closed sets V q (I) where I is any ideal of R. Since any primary ideal is quasi-primary, we have P rim(R) as a subspace of QP rim(R).
For the sake of completeness, we note some properties (from Proposition 2.2 to Corollary 2.6) of V q (I) without proofs. For details, see [5].
Proposition 2.2. Let I, J be ideals of R and {I λ } λ∈Λ a family of ideals of R. Then the followings hold: This topology is called Zariski topology on QP rim(R) and the space QP rim(R) is named as the quasi-primary spectrum of R. We note that any open set in QP rim(R) is of the form QP rim(R)\V q (S) for some subset S of R.
Set U a = QP rim(R)\V q (a) for any a ∈ R.
Theorem 2.4. Let R be a ring. The family {U a } a∈R is a base for the Zariski topology on QP rim(R).
Note that U 0 = ∅ and U r = QP rim(R) for every unit r ∈ R.
Theorem 2.5. Let R be a ring and a, b ∈ R. The followings hold: Quasi-primary ideals firstly introduced and examined thoroughly in [3]. It is generally studied on rings satisfying maximal condition; in other words, that every ascending chain of ideals is finite. It is also noted that the quasi-primary ideals in rings satisfying maximal condition can be characterized as follows: A quasi-primary ideal is either a power of a prime ideal or an intermediate ideal between two powers of one and the same prime ideal. In the view of this fact, the following theorem is given for rings satisfying maximal condition.
Theorem 2.7. [3, Theorem 4] If Q 1 and Q 2 are quasi-primary ideals having the radicals P 1 and P 2 respectively, and P 1 ⊆ P 2 . Then Q 1 Q 2 is also quasi-primary having the radical P 1 .
Theorem 2.8. Let R be a ring satisfying maximal condition and Q 1 and Proof. It follows from Theorem 2.7.
It is known that Theorem 2.7 has no analogue in primary ideal theory. Similarly, Theorem 2.8 does not valid for the primary spectrum as can be seen in the following example.
Example 1. Consider the residue class ring R = K[X 1 , X 2 , X 3 ]/(X 1 X 3 − X 2 2 ) where K is a field. It is clear that R satisfies maximal condition. Let x i denote the natural image of X i in R for each i = 1, 2, 3. Then, the ideal P = (x 1 , x 2 ) is a prime ideal of R but P 2 is not primary [7,Example 4.12]. It is trivial that P 2 is a quasi-primary ideal of R. Now take Q 1 = Q 2 = P. Then, we can see that P ∈ V q (P ) ∩ V rad (P ), however P 2 ∈ V q (P ) \ V rad (P ). Now, let us determine the closure of a point Q ∈ QP rim(R). The closure . Then, we obtain N (R) ∈ U . In a similar way, we get N (R) ∈ V . Consequently, U ∩ V = ∅. For the other part, let QP rim(R) be an irreducible space. Assume that N (R) is not quasi-primary. Then N (R) is not prime. Then there exist a, b ∈ R such that a, b / ∈ N (R) but ab ∈ N (R). Since a ∈ N (R), V q (a) = QP rim(R), that is, There is a one to one correspondence between points of QP rim(R) and irreducible closed subsets of QP rim(R). The next theorem gives that correspondence. Proof. Let Y = V q (Q) for any Q ∈ QP rim(R). Since V q (Q) = Cl(Q) and Cl(Q) is irreducible, Y is an irreducible closed subset of QP rim(R). Conversely, let Y be an irreducible closed subset of QP rim(R). Then Y = V q (I) for some ideal I of R. Now suppose that I / ∈ QP rim(R). Then √ I is not prime. Then there are elements a, b ∈ R such that ab ∈ I but a, b / ∈ √ I. Thus, So we obtain Y reducible which contradicts our assumption.
Let I be an ideal of a ring satisfying maximal condition. Then, by [3, Theorem 5], the ideal I is the intersection of a finite number of quasi-primary ideals, say Q 1 , . . . , Q n with radicals P 1 , . . . , P n , respectively. Hence, there is no prime ideal containing I other than P i 's where i = 1, . . . , n. Then, for any ideal I in a ring satisfying maximal condition, every closed subset V q (I) can be written as the finite union of irreducible closed sets, that is, V q (I) = V q (P 1 ) ∪ · · · ∪ V q (P n ) by Proposition 2.2 (iii) and (v).
Let V be a closed subset of a topological space X. A dense point of V is called a generic point. By the above theorem, we conclude that every irreducible closed subset of QP rim(R) has a generic point.
The maximal irreducible subsets of a topological space X are called irreducible components.
Proof. By Theorem 2.11, any irreducible closed subset of QP rim(R) can be written of the form V q (Q) for some quasi-primary ideal Q of R.
The following lemma is easy to prove, so we left it as an exercise.
Theorem 2.14. Let R be a ring. The following are equivalent: Then we have I + J = R and I ∩ J = IJ. So, we get R = R/I × R/J. (ii) =⇒ (iii) Assume that R ∼ = R 2 × R 2 where R 1 and R 2 are nonzero rings via an isomorphism φ. Then φ −1 (1, 0) is a nontrivial idempotent of R.
The dimension of a topological space X is the number n such that X has a chain of irreducible closed sets V 1 ⊂ V 2 ⊂ · · · ⊂ V n and no such chain more that n terms.
Theorem 2.15. Let R be a Noetherian local ring. Then the dimension of QP rim(R) is finite.
be a chain of irreducible subsets. This chain can be written as where Q i ∈ QP rim(R). Let P i = √ Q i for each i. Then we have · · · ⊂ P n ⊂ · · · ⊂ P 2 ⊂ P 1 .
By [4,Corollary 11.11], the dimension of R is finite. So the above chain of prime ideals must terminate. Therefore the dimension of QP rim(R) is finite, and in fact equal to the dimension of R.

A Sheaf of Rings on the Quasi-Primary Spectrum
In this section we define a sheaf of rings on the quasi-primary spectrum. Let φ : R → R ′ be a ring homomorphism. For any Q ∈ QP rim(R ′ ), it is easy to show that f −1 (Q) ∈ QP rim(R). So f induces the map which is called the associated map of φ.
For any A ⊆ R we have (φ a ) −1 (V (A)) = V (φ(A)). So the map φ a is continuous. Let S ⊆ R be a multiplicative subset of R. Let φ : R → R S be the canonical homomorphism. Since √ IR S = √ IR S for any ideal I of R, the map φ a is an inclusion. The set U S = φ a (QP rim(R S )) is equal to the set of quasi-primary ideals of R whose radicals are disjoint from S. There is a one-to-one correspondence between quasi-primary ideals of R S and quasi-primary ideals of R whose radicals are disjoint from S. So, the space QP rim(R S ) is homeomorphic to the subspace U S of QP rim(R). In Proof. Assume that U a ⊆ U b for some a, b ∈ R. Then, for any Q ∈ QP rim(R), Since QP rim(R) contains prime ideals, this observation yields that a ∈ (b). Conversely, assume that a ∈ (b) for some a, b ∈ R. Let q ∈ U a . Then a ∈ √ Q. Since a is contained in the intersection of all prime ideals that contain b, we obtain that b ∈ √ Q. Therefore, we have Q ∈ U b .
Our aim is to construct a sheaf of rings on QP rim(R). We assign to each open set U a the ring F (U a ) := R a , ring of quotients with respect to the multiplicative subset {1, a, a 2 , ...}, and define the restriction maps Since, by Lemma 3.1, we have U a ⊆ U b if and only if a n = tb for some positive integer n and t ∈ R, the map res U b ,Ua is well-defined.
For an arbitrary open set U of QP rim(R) let where the projective limit is taken over all U a ⊆ U relative to the system of homo- With this construction, F turns to be a sheaf of rings on QP rim(R). In fact, this sheaf is the direct image sheaf under the inclusion map from Spec(R) into QP rim(R): Proof. The inclusion map ι is continuous. For any open set U of QP rim(R), direct image sheaf ι * is defined as follows: where O denotes the structure sheaf on Spec(R). For a ∈ R, we have ι −1 (U a ) = {P ∈ SpecR : ι(P ) ∈ U a } = {P ∈ SpecR : P ∈ U a } = {P ∈ SpecR : a ∈ √ P = P } The final set is a principal open set for Spec(R) and the corresponding ring for this set is R a . So we get ι * (U a ) = R a = F (U a ). For U a ⊆ U b , we have res U b ,Ua = ρ X b Xa where ρ X b Xa is the restriction map from principal open set X b to X a of Spec(R) with respect to the structure sheaf O. Thus the sheafs F and ι * F are the same.
Similar to the structure sheaf O on Spec(R), the stalk F Q of F at a point Q ∈ QP rim(R) is R √ Q . Therefore, we conclude that (QP rim(R), F ) is a locally ringed space. Finally, since F is the direct image sheaf of O under the inclusion map ι : Spec(R) → QP rim(R), it is easily seen that (QP rim(R), F ) is a scheme.