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Chapter 2 Quasi m-spaces and quasi cozero complemented frames
Introduction
We denote the annihilator of a set S by Ann(S), and abbreviate Ann(fag) as Ann(a). Double annihilators are written as Ann2(S). An ideal I of a ring A is called a d-ideal if Ann2(a) I, for every a 2 I. On the other hand, I is called a z-ideal if whenever a 2 I and b is an element of A contained in every maximal ideal containing a, then b 2 I. The symbols Spec(A), Max(A) and Min(A) have their usual meaning; namely, the set of prime, maximal and minimal prime ideals of A, respectively. We write Specd(A) and Specz(A) for the set of prime d-ideals and prime z-ideals of A, respectively.
One of our goals is to characterize frames L such that, for each of these conditions, the ring RL satis es that condition. In the class of reduced rings, three of these are all equivalent, and are equivalent to the property of being von Neumann regular (vNR). Precisely, zMin () zMax () dMax () vNR: Indeed, (zMin) is equivalent to (vNR) because maximal ideals are z-ideals, and a reduced ring is von Neumann regular if and only if every maximal ideal is minimal prime. Next, (zMax) implies (dMax) because every d-ideal is a z-ideal([55, Proposition 2.12]); (dMax) implies (vNR) because every minimal prime ideal is a d-ideal; and, nally, (vNR) implies (zMax) because every prime ideal in a von Neumann regular ring is a maximal ideal. Thus, RL satis es any (and hence all) of these three if and only if L is a P -frame because L is a P -frame if and only if RL is von Neumann regular [12].
We shall see that RL satis es (dMin) precisely if L is cozero complemented. The more substantive results concern those L for which RL satis es (dMM).
Characterizations of quasi cozero complemented frames
In this section we proceed to characterize frames L for which every prime d-ideal of RL is either a maximal ideal or a minimal prime ideal. But rst we start by justifying the claim made in the introduction that every prime d-ideal of RL is minimal prime if and only if L is cozero complemented. Let us recall the de nition. A frame L is cozero complemented if for every c 2 Coz L there is a d 2 Coz L such that c _ d is dense and c ^ d = 0.
We recall that a ring R is said to have Property A if every nitely generated ideal of R which consists entirely of zero-divisors has nonzero annihilator (see, for instance, [42]). In [2, Proposition 1.26] the authors show, among other things, that if a ring R has Property A, then every prime d-ideal of R is minimal prime if and only if for every a 2 R there exists b 2 R such that Ann(a) = Ann2(b). Proposition 1.1 in [32] shows that L is cozero complemented if and only if for every 2 RL there is a 2 RL such that Ann( ) = Ann2( ). Now let us show that RL has Property A.
lemma 2.2.1. The ring RL has property A.
Proof. In fact, let Q = h 1; 2 mi be a nitely generated ideal of RL consisting of zero-divisors. Then 12 + 22 + + m2 2 Q and therefore a zero-divisor. Note that fcoz j 2 Qg = coz( 12) + coz( 22) ++ coz( m2); which is not dense because 12 + 22 + + m2 is a zero divisor. Thus by [24, Lemma 4.3], Ann(Q) 6= 0.
We therefore have the following proposition.
Proposition 2.2.2. Every prime d-ideal of RL is minimal prime if and only if L is cozero complemented.
Remark 2.2.3. There is an alternative a rmation of this result. It is shown in [29, Proposition 3.1] that the lattice Did(RL) of d-ideals of RL is a coherent frame, and, is in fact, the frame of d-elements of Rad(RL), where the latter denotes the frame of radical ideals of RL. Now, by [52], every prime d-ideal of RL is minimal prime if and only if Did(RL) is regular. This in turn is equivalent to L being cozero complemented in light of [29, proposition 5.5]. In fact an algebraic frame is regular if and only if every compact element is complemented, and the compact elements of Did(RL) are precisely the ideals Mc , for c 2 Coz L ([29, Proposition 4.1]).
Now we investigate when every prime d-ideal in RL is either a maximal or a minimal prime ideal. We rst obtain a characterization for reduced f-rings. It will generalize the equivalence of conditions (i) and (ii) in [1, Theorem 3.2]. We start with a lemma which is itself an f-ring version of [1, Lemma 3.1]. Observe that a directed union of d-ideals is a d-ideal.
In the upcoming proof we are going to use the fact that a prime ideal minimal over a d-ideal is itself a d-ideal [55, Theorem 2.5]. We shall also have to keep in mind that a prime ideal P in a reduced ring is minimal prime if and only if, for every a 2 P , there exists b 2= P such that ab = 0 [39].
Proposition 2.2.5. The following are equivalent for a reduced f-ring A.
(1)Every prime d-ideal of A is either a maximal ideal or a minimal prime ideal.
(2) For every maximal ideal M of A and every pair a; b of elements in M; there exists u 2 Ann(a) and v 2= M such that Ann(a2 + u2) Ann2(bv).
Proof. (1) ) (2): Suppose (2) fails. Then there is a maximal ideal M of A and elements a; b 2 M such that for every u 2 Ann(a) and v 2= M, Ann(a2 + u2) * Ann(bv). De ne subsets I and S of A by [I =fAnn2(a2 + t2) j ta = 0g and S = fbnr j r 2= M and n = 0; 1; : : :g:
It is easy to check that S is multiplicatively closed because M is a prime ideal. We claim that S \I = ;. If not, then bnr 2 I for some n and r 2= M, and hence bnr 2 Ann2(a2 + t2) for some t 2 Ann(a). Thus, hbnri Ann2(a2 + t2), which implies
Ann(a2 + t2) = Ann(Ann2(a2 + t2)) Ann(bnr) = Ann(br);because A is reduced. But this violates the supposition. There is therefore a prime ideal P which contains I and misses S. Without loss of generality, we may assume P is minimal with this property. Then P is a d-ideal ([55, Theorem 2.5]). Clearly, ArM S, so that, in light of P \ S = ;, we have P M. Observe that a 2 P because a 2 Ann2(a) I P . Also, Ann(a) P because of the following. If ta = 0, then Ann2(a2 + t2) I P .
Consider any z 2 A with z(a2 + t2) = 0. Then z2a2 = z2t2 = 0, whence tz = 0. Thus, t 2 Ann2(a2 +t2) P . It follows therefore that P is not a minimal prime ideal. Then P is a maximal ideal by (1), and hence P = M. But b 2 M \ S = P \ S = ;, a contradiction.
(2)) (1): Let P be a prime d-ideal and M a maximal ideal with P M. Suppose, for contradiction, that P 6= M and P is not a minimal prime ideal. Since P is properly contained in M, there exists b 2 M r P . Because P is not a minimal prime ideal, there is an a 2 A such that a 2 P and Ann(a) P . By (2), there exist u 2 Ann(a) and v 2= M
such that Ann(a2 + u2) Ann(bv). Since a2 + u2 2 P and P is a d-ideal, it follows that bv 2 P . Since b 2= P , this implies v 2 P M, which is a contradiction.
De nition 2.2.6. A ring is a quasi m-ring if every prime d-ideal in it is either maximal or minimal prime. If RL is a quasi m-ring, we shall say L is a quasi cozero complemented frame.
In light of Proposition 2.2.2, every cozero complemented frame is a quasi cozero com-plemented frame. We shall shortly give a frame-theoretic characterization of quasi cozero complemented frames. Let us recall some facts from [26]. For any 2 RL,
1 Introduction and preliminaries
1.1 A brief history on P-spaces and P-frames
1.2 Synopsis of the thesis
1.3 Frames and their homomorphisms
1.4 Rings and f-rings
1.5 Function rings
1.6 The coreflections λL and υL
1.7 Binary coproducts of frames
2 Quasi m-spaces and quasi cozero complemented frames
2.1 Introduction
2.2 Characterizations of quasi cozero complemented frames
2.3 Subspaces of quasi m-spaces
3 Quasi P-frames
3.1 Characterizations of quasi P-frames
3.2 Some special quasi P-frames
4 Weak almost P-frames
4.1 Introduction
4.2 Characterizations of weak almost P-frames
4.3 More on weakly regular f-rings
5 Boundary frames
5.1 Introduction
5.2 A ring theoretic characterization of boundary frames
5.3 On product of boundary spaces
5.4 Some comments on boundary rings
6 Frames that are finitely an F-frame
6.1 Introduction
6.2 Ring theoretic characterization
6.3 Inheritance by quotients
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VARIANTS OF P-FRAMES AND ASSOCIATED RINGS
