On 2 r - ideals in commutative rings with zero - divisors

: In this article, we are interested in uniformly pr - ideals with order ≤ 2 ( which we call r 2 - ideals ) introduced by Rabia Üregen in [ On uniformly pr - ideals in commutative rings , Turkish J. Math. 43 ( 2019 ) , no. 4, 18781886 ] . Several characterizations and properties of these ideals are given. Moreover, the comparison between the ( nonzero ) r 2 - ideals and certain classes of classical ideals gives rise to characterizations of certain rings based only on the properties of the ideals consisting only of zero - divisors. Namely, among other things, we compare the class of ( nonzero ) r 2 - ideals with the class of ( minimal ) prime ideals, the class of minimal prime ideals and their squares, and the class of primary ideals. The study of r 2 - ideal in poly - nomial rings allows us to give a new characterization of the rings satisfying the famous A - property.


Introduction
Throughout, all rings considered are commutative with nonzero unity. Let R be a ring, I be an ideal of R, and S be a subset of R. Set . The ring R is said to be a total quotient ring if ( ) = R Q R , or equivalently, every element in R is either a zero-divisor or a unit.
It is known that there are so many important rings with zero divisors that have interesting properties whose counterparts for the integral domains become trivial. Recently, there has been a lot of attention to the ideal theory of these rings (see [1][2][3][4]). For a ring R, the properties of ( ) Q R provided by its ideals come from the ideals of R consisting entirely of zero divisors. An example of such ideals are the z 0 -ideals (studied under the name d-ideals in [4]). Definition 1.1. [1] A proper ideal I of a ring R is said to be a z 0 -ideal if, for each ∈ a I, we have ⊆ P I a , where P a is the intersection of all minimal prime ideals containing a and, by convention, = P R a if a is not contained in any minimal prime ideal.
If R is reduced, then ( ) Q R is Von Neumann regular if and only if every proper (principal) ideal is a z 0 -ideal [1,Corollary 1.14]. In [5], Mohamadian generalized the notion of ideals z 0 by introducing the concepts of r-ideals. Definition 1.2. [5] A proper ideal I of a ring R is called an r-ideal if, whenever, ∈ x y R , with ∈ xy I , we have ∈ x I or ( ) ∈ y R Z , which are also consisting entirely of zero divisors.
Let ( ) C X be the ring of real valued continuous functions on a Tychonoff space X. It is proved in [5,Proposition 5.4] that, over ( ) C X , every r-ideal is a z 0 -ideal if and only if X is a ∂-space (a space in which the boundary of any zeroset is contained in a zeroset with empty interior). Several recently introduced notions are related to the notion of r-ideals (see, for example, [6,7]).
In [8], Uregen introduced the concept of uniformly pr-ideals as follows: A proper ideal I of a ring R is said to be a uniformly pr-ideal if there exists a positive integer n such that, whenever ∈ x y R , with ∈ xy I , then ∈ x I n or ( ) ∈ y Z R . The order of I is the smallest positive integer for which the aforementioned property holds.
Hence, r-ideals are just uniformly pr-ideals with order 1. So, the order of an uniformly a pr-ideal I measures how far away I is from being an r-ideal.
In this article, we are interested to uniformly pr-ideals with order ≤2, which we call r 2 -ideals. Section 2 gives several characterizations and properties of r 2 -ideals. It is also proved that every r-ideal is a r 2 -ideal and that minimal prime ideals and their squares are r 2 -ideals. Thus, several results in Section 2 study the rings in which every r 2 -ideal is an r-ideal, the rings in which the only (nonzero) r 2 -ideals are the minimal prime ideals (resp. and their squares), and the rings in which every (nonzero) r 2 -ideals are primary. These comparisons give rise to characterizations of several rings using the properties of ideals consisting of zerodivisors. Section 3, among other things, studies some types of r 2 -ideals in the polynomial rings. In addition, the r 2 -ideals are used to characterize the rings satisfying the A-property.

On r -ideals in commutative rings
In this section, we introduce the class of r 2 -ideals and study some of their properties. We also compare the class of r 2 -ideals with some other classes of ideals.
By definition, the r-ideals are r 2 -ideals and the r 2 -ideals are uniformly pr-ideals. However, these concepts are different, as the following example shows.  are regular elements of R. It is also easy to see that ( ) 1 2 , and ( ) + ∉ x y Z R . Accordingly, ( ) P 1 2 is not an r-ideal. The ideal ( ) P 1 3 is a uniformly pr-ideal (with order 3). On the other hand, ( ) ( ) Recall that a proper ideal I of a ring R is said to be strongly quasi primary if, whenever, ∈ a b R , with ∈ ab I , then either ∈ a I 2 or ∈ b I [9]. Strongly quasi primary ideals contain primary ideals, and they are used to characterize divided domains [9, Theorem 2.2].
The first result in this section, while an immediate consequence of the definition of r 2 -ideals, is an important fact since it emphasizes that r 2 -ideals are entirely consisting of zero-divisors.
Proposition 2.3. Let R be a ring and I be a proper ideal of R. If I is a r 2 -ideal, then ( ) ⊆ I R Z . The equivalence holds if I is strongly quasi primary.
Proof. For each ∈ a I, we have ∈ a I 1.
and ∉ I 1 2 . Then, ( ) ∈ a Z R , and so ( ) Recall that, for a proper ideal I of a ring R, the ideal generated by the squares of elements of . Therefore, ∈ xr I for some Following [5, Proposition 2.2], the r-ideals of a ring R are exactly the contraction of the proper ideals of ( ) Q R . Similarly, the aforementioned theorem shows that the r 2 -ideals are the ideals lying between J c and [ ] J c 2 , where J ranges over all the proper ideals of ( ) Q R . The following theorem gathers some useful facts about r 2 -ideals. These facts will be used in the sequel without explicit mention.
Theorem 2.5. Let R be a ring. Then, the following hold. 1. If I is a r 2 -ideal of R, then I is an r-ideal. 2. If I is an r-ideal of R and J is an ideal of R such that [ ] ⊆ ⊆ I J I 2 , then J is a r 2 -ideal. In particular, I 2 and [ ] I 2 are r 2 -ideals. 3. If I is an r-ideal of R and ⊆ I J is an ideal of R, then IJ is a r 2 -ideal. 4. If P is a minimal prime ideal of R, then P is an r-ideal and so P 2 and [ ] P 2 are r 2 -ideals. 5. If I and J are r 2 -ideals (resp. r-ideals) of R, then ∩ I J is a r 2 -ideal (resp. an r-ideal).
6. If I and J are r-ideals of R, then IJ is a r 2 -ideals. 7. Let I and J be coprime ideals (I and J are proper and + = I J R). If = ∩ IJ I J is a r 2 -ideal, then so are I and J. 8. Every maximal r 2 -ideal of R is prime. 9. If I is a r 2 -ideal of R and P is a minimal prime ideal over I, then P is an r-ideal. Proof.
(1) Clear. . . Hence, , with ∈ ab P and ∉ a P. We have to show that ∈ b P. By (1), P is an r-ideal, and so a r 2 -ideal. By the maximality of P, we obtain = P P . Thus, P is an r-ideal. It is easy to see that ( ) P a : is proper and is an r-ideal, which contains of course P. Again, by the maximality of P, we obtain that ( ) = P P a : . Hence, .
y P i , a contradiction. Hence, + x y does not belong to any minimal prime ideal, and then + x y is regular. Let J be an ideal of R containing Q j . Since Q j is an r-ideal (as an intersection of r-ideals), Q J j is a r 2 -ideal (by Theorem 2.5(3)), and so it is an r-ideal. Since ( ) where I i is an ideal of D i for each i. Since J must be proper, there exists some ideals I i , which are proper (in D i ). Suppose, for example, that ≠ I R . Consequently, each I i in J is either ( ) 0 or equal to D i . Thus, = J J. Hence, by Theorem 2.5 (1), J is an r-ideal, as desired. □ Of course, a ring need not be a finite direct product of domains to have the property that every r 2 -ideal is an r-ideal. In fact, we can consider any total quotient ring, and so in such a ring, any ideal is an r-ideal [5].
The following result shows that if every ideal is a r 2 -ideal, then R is also a total quotient ring.
Theorem 2.7. Let R be a ring. Then, the followings are equivalent: 1. R is a total quotient ring.

Every proper ideal is an
Recall that a proper ideal I of a ring R is said to be semi-primary if I is prime. Recall also that a ring is said to be decomposable if it admits a nontrivial idempotent.
Lemma 2.8. Let R be a ring such that every nonzero r 2 -ideal is semi-primary. Then, 1. either ( ) ( ) ≠ R nil 0 and ( ) Z R are prime ideals (and so R is indecomposable), 2. or R is reduced with at most two minimal prime ideals.
Proof. Assume that R is not reduced. Let P and Q be two prime ideals contained in ( ) Z R . Since R is not reduced, ( ) ∩ ≠ P Q 0 . Since ∩ P Q is an r-ideal, we obtain that ∩ = ∩ P Q P Q is prime. Hence, P and Q are comparable. In particular, minimal prime ideals are comparable. Hence, ( ) = P nil R is prime. On the other hand, ( ) Z R is a union of some prime ideals (which are necessarily contained in ( ) Z R , and so they are comparable). Hence, ( ) Z R is an ideal, and so prime. Suppose that R is decomposable, and let e be a nontrivial idempotent of R. Then, , a contradiction. Suppose now that R is reduced but not a domain. Let P 1 and P 2 be two different minimal prime ideals.
, then, as earlier, P 1 and P 2 are comparable, a contradiction. Then, ( ) ∩ = P P 0 1 2 , and so R admits exactly two minimal prime ideals. □ Lemma 2.9. Let R be a ring such that every r 2 -ideal is semi-primary. Then, is prime. Now, the result follows from Lemma 2.8. □ When a prime ideal consists entirely of zero-divisors, then it must be an r-ideal (and so a r 2 -ideal). The next proposition shows that the r 2 -ideals are not necessarily all prime, except in a domain.
Proposition 2.10. Let R be a ring. Then, the followings are equivalent: is an r-ideal, and so a r 2 -ideal. Thus, The fact that ( ) 0 is an r-ideal forces the ring in Proposition 2.10 to be a domain. To avoid this, we will now focus on the noetherian rings in which only the nonzero r 2 -ideals must be (minimal) prime. Recall that a minimal ideal of a ring R is a nonzero ideal that does not contain any other nonzero ideal of R.
Theorem 2.11. Let R be a noetherian ring. Then, the following are equivalent: , and so ( ) ( ) = Z R e is generated by an idempotent e.
Hence, by Theorem 2.5(2), I is a r 2 -ideal, and then prime. Since Z R is a minimal ideal. Assume now that R is reduced. Since every nonzero r 2 -ideal is a prime ideal contained in ( ) Z R , we conclude that every (nonzero) r 2 -ideal is an r-ideal. Hence, from Theorem 2.6, R is a finite direct product of domains. On the other hand, by Lemma 2.8, R admits at most two minimal prime ideals. Thus, R is a domain or is isomorphic to a product of two domains.
( ) ( ) ⇒ 3 1 If R is a domain, the conclusion is obvious. Now, suppose that ≅ × R D D 1 2 for some domains D 1 and D 2 . As in the proof of Theorem 2.6, the only nonzero (a) R is a domain, nil is the unique minimal prime ideal of R. By Lemma 2.9, ( ) Z R is an ideal, and so a nonzero r-ideal. Hence, ( ) = Z R P. Let I be a nonzero subideal of ( ) Z R . We have ( ) = ⊆ ⊆ P I P 0 2 . Hence, I is a r 2 -ideal, and then ( ) 1 If R is a domain, then the result is clear. So, suppose that ( ) Z R is a minimal ideal. Then, ( ) Z R is the unique minimal prime ideal of R (since ( ) 0 is not prime). Every nonzero r 2 -ideal I is a subideal of ( ) Z R , and so ( ) = I Z R . Hence, it suffices to show that ( ) 0 is a square of ( ) Z R . Let ( ) ∈ * a Z R . By the minimality of ( ) . Thus, = a a r 2 for some ∈ r R. Since ( ) ( ) ∈ = a Z R R nil , we obtain that = a 0, a contradiction. So, again by the minimality of ( ) The next three results investigate when, in noetherian rings and decomposable rings, the only nonzero r 2 -ideals are the minimal prime ideals and their squares.
Theorem 2.13. Let R be a decomposable ring. Then, the following are equivalent: 1. The only nonzero r 2 -ideals are P and P 2 with ( ) ∈ P R Min . 2. Every nonzero r 2 -ideal is primary.

Every nonzero r
2 -ideal is semi-primary. 4. R is isomorphic to a product of two domains.

Every nonzero r
2 -ideal is a minimal prime.
4 By Lemma 2.8, since R is decomposable, R must be reduced and it admits exactly two minimal prime ideals P 1 and P 2 (since a domain is indecomposable). Let e be a nontrivial idempotent of R. Either ∈ e P 1 or ∈ e P 2 . Suppose, for example, that ∈ e P 1 . Then, − ∈ e P 1 2 . Hence, + = P P R 1 2 , and so ≅ / × / R R P R P , ( ) Z R 2 is a minimal ideal, and there is no ideal strictly between ( ) Z R and ( ) Z R 2 .
, then every nonzero subideal of P is a r 2 -ideal (by Theorem 2.5(2)), and so equal to P. Thus, ( ) = P Z R is a minimal ideal. Now, assume that ( ) ≠ P 0 2 . Suppose that P 2 is not an r-ideal. Hence, there exists ∉ x P 2 and ( ) ∉ = r Z R P such that ∈ xr P 2 . Clearly, ∈ x P. Now, we have ( ) is a r 2 -ideal, and so ( ) . Thus, for each ∈ p P, = + p xa b for some ∈ a R and ∈ b P 2 . Hence, = + ∈ pr xra br P 2 . So, ⊆ rP P 2 . Since R is noetherian, there exists an integer ≥ n 3 such that 2 , I is a r 2 -ideal. So, = I P or = I P 2 . If = I P then = P P 2 , and so ( ) = P 0 , a contradiction. Thus, = I P 2 , and so P 2 is a minimal ideal. Moreover, each ideal between P and P 2 is a r 2 -ideal, and then equal to P or P 2 . Thus, there are no ideals strictly between P and P 2 .
( ) ( ) ⇒ 2 1 It is clear that, in the both cases, must be the unique minimal prime ideal of R. Suppose that ( ) Z R is a minimal ideal and let I be a nonzero r , and then, . Hence, ⊊ ⊆ P I P 2 , and so = I P. □ It easy to verify that every primary ideal consisting of zero divisors is an r-ideal, and so a r 2 -ideal. In the rest of this section, we are interesting to rings in which every (nonzero) r 2 -ideal of R is a primary. Recall from [12] that a ring R is said to be an UN -ring if every nonunit element a of R is a product of a unit and nilpotent elements. Following [  for some ∈ r R. Thus, nil , a contradiction. So, R must be either domain or a UN -ring.
( ) ( ) ⇒ 2 1 If R is a domain, then ( ) 0 is the only r 2 -ideal of R, which is primary. If R is a UN -ring. Then, every proper ideal of R is primary, in particular every r 2 -ideal of R is primary. □ Theorem 2.17. Let R be a noetherian ring. Then, the following are equivalent: 1. Every nonzero r 2 -ideal of R is primary. 2. One of the following holds: (a) R is a domain.
is prime and minimal, ( ) Z R is a maximal ideal, and there are no prime ideals strictly between ( ) R nil and ( ) Z R .
Proof. ( ) ⇒ Suppose that R is reduced. Let I be a nonzero r 2 -ideal. Then, ( ) ⊆ I Z R , and so ( ) Since I is primary, we obtain that ∈ a I. Hence, I is an r-ideal. Thus, every (nonzero) r 2 -ideal is an r-ideal. By using Theorem 2.6, we conclude that R is isomorphic to a finite product of domains. by Lemma 2.8, R admits at most two minimal prime ideals. Thus, R is a domain or is isomorphic to a product of two domains.
Suppose that R is not reduced. By Lemma 2.8, Thus, , and there is no ideal strictly between P and ( ) Z R . Let ( ) = ≠ ⊆ P I P 0 2 . Then, I is a r 2 -ideal, and so primary. Let ∈ x P and ( ) ∈ z Z R P \ . We have that ( ) = + ⊆ J I xz P, and it is also primary. We have ( ) ∈ + xz I xz , and , we obtain that ∈ x I. Thus, = I P. So, P is a minimal ideal of R. Consider a maximal ideal M such that ( ) ⊆ Z R M. We have ( ) = MP 0 or = MP P (Since P is minimal). On the other hand, by minimality of P, we obtain that it is principal. So set = P xR. If  I R nil , which is prime. In the second case, , be a local noetherian ring. Then, the followings are equivalent: 1. Every nonzero r 2 -ideal of R is primary. 2. One of the following holds: is a minimal ideal of R.

r 2 -Ideals in polynomial rings
We begin this section with following easy fact.   .
Assume that ≠ f 0. . We have: Recall that a ring R is said to be an A-ring if each finitely generated ideal contained in ( ) R Z has a nonzero annihilator. It is well known that noetherian rings, zero-dimensional rings, and polynomial rings are A-rings. Moreover, a ring R is an A-rings if and only if so is ( ) Q R (see [11,15,16]). Next, we characterize the A-rings in terms of r 2 -ideals.
3 Theorem 2.7 states that every proper ideal of ( ) Q R is a r 2 -ideal. Moreover, R is an A-ring if and only if so is ( ) Q R . Now, applying the equivalence ( ) ( ) ⇔ 1 2 for ( ) Q R , we obtain that ( ) Q R is an A-ring if and only if for each proper ideal J of ( ) We now turn to characterize the r 2 -ideals of the quotient rings.
Proposition 3.5. Let ⊆ I J be two ideal of a ring R. The followings are equivalent: .
In particular, if I is an r-ideal of R and J/I is a r 2 -ideal of R/I, then J is a r 2 -ideal of R. .
The next example shows that even if I is an r-ideal of R and J is a r 2 -ideal of R, then / J I needs not to be a r 2 -ideal of / R I .
. Suppose Q is a prime ideal of A P that is not a maximal. Then, there exists a prime ideal q of A that is contained strictly in P such that = Q qA P . Moreover, we can write q as is a chain of prime ideals, we conclude that ( ) = q x 0 . Therefore, ( ) = / Q x 1 . So, R has exactly one nonmaximal prime ideal, which is and M are r-ideals of R. However, 0 is not a r 2 -ideal of the domain / R P (by Proposition 2.10).
be a ring homomorphism. The following are equivalent: , and thus, Reg . □ Corollary 3.8.
• Let ⊆ R T be rings such that R is essential in T (i.e., ( ) ∩ ≠ R I 0 for every nonzero ideal I of T). If I is a r . It is also called the (Nagata) idealization of M over R and is denoted by ( ) + R M. This construction was first introduced, in 1962, by Nagata [18] with the objective to emphasize the interaction between rings and their modules and, more importantly, to provide numerous families of examples of rings with zero-divisors.