Reticulated rings

Simmons, Harold

doi:10.1016/0021-8693(80)90118-0

Cited by 68 publications

(60 citation statements)

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“…As an application of the above results we prove a result that was proved by Simmons [9]. Speed [10] proved that a lattice L is quasi complemented if and only if the space of minimal prime ideals with the hull-kernel topology is compact.…”

Section: Theorem 7 a Semiprime Ring R Is A Pf-ring If And Only If Lrmentioning

confidence: 84%

“…Hence D : ~R ~ ~LR is a lattice isomorphism. So, by Theorems 1, 5, and 6, we get the following theorem that answers a question raised by Simmons [9].…”

Section: Theorem 6 a Lattice L Is Conormal If And Only If Each Minimmentioning

confidence: 87%

“…In this paper we continue this theme and answer a couple of questions raised by Simmons [9]. Then we proceed to prove further results in that direction.…”

Section: D(1r)=i D(0r)=0 D(ab) = D(a) a D(b) D(a + B) <= D(a) V Dmentioning

confidence: 96%

“…For more details about the reticulation LR of a ring, see Simmons [9]. The reticulation of a ring was investigated by Simmons [9] in order to show that alot of ring theoretic properties have analogues in lattice theory and vice versa.…”

Section: D(1r)=i D(0r)=0 D(ab) = D(a) a D(b) D(a + B) <= D(a) V Dmentioning

confidence: 99%

“…For any ideal J of the lattice LR, let D-l(J) = {a E R: D(a) E J). Trivially, D-I(J) is an ideal of R.The reticulation of a ring was investigated by Simmons [9] in order to show that alot of ring theoretic properties have analogues in lattice theory and vice versa. In this paper we continue this theme and answer a couple of questions raised by Simmons [9].…”

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confidence: 99%

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Further results on reticulated rings

Al-Ezeh

1992

Acta Math Hung

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Throughout this paper all rings are commutative with unity and all lattices are distributive with 0 and 1 unless otherwise stated. For a ring R, the reticulation of R is defined as a distributive lattice generated by the symbols D(a), a E R and satisfying: D(1R)=I, D(0R)=0, D(a.b) = D(a) A D(b), D(a + b) <= D(a) V D(b).For more details about the reticulation LR of a ring, see Simmons [9]. For any ideal I of the ring R, let D(I) be the ideal generated by {D(a) : a E I) in LR. For any ideal J of the lattice LR, let D-l(J) = {a E R: D(a) E J). Trivially, D-I(J) is an ideal of R.The reticulation of a ring was investigated by Simmons [9] in order to show that alot of ring theoretic properties have analogues in lattice theory and vice versa. In this paper we continue this theme and answer a couple of questions raised by Simmons [9]. Then we proceed to prove further results in that direction. Let Id(R) be the lattice of all ideals of the ring R. It should be noted that this lattice is not necessarily distributive. Let RId(R) be the distributive lattice of radical ideals of the ring R. For a lattice L, let Id(L) be the lattice of ideals of L. From now on, LR will denote the reticulation of R.We start by quoting a theorem that was given by Johnstone [7], p. 194. THEOREM 1. Let R be a ring. Then D -1 : Id(LR) --* RId(R) is a lattice isomorphism, moreover for any ideal I of the ring R, D-I(D(I)) = Vf[, the radical of I. Also, D -t defines a bijection from the prime ideals of LR to the prime ideals of R. Hence minimal prime ideals of LR correspond to minimal prime ideals of R.Recall that a ring R is called quasi regular if for every a E R, there exists b e R such that ann(ann(a)) = ann(b), where for any ideal I of R, ann(I) = {x e R:xy = 0 for all y E I). Alattice Lis called quasi complemented if for every a E L, there exists b E L such that a** = b*, where for any ideal I of a lattice L, I* = {x E L: x A y = 0 for all y E I). The first result we want to establish is the following: A semiprime ring (without nontrivial nilpotents) is quasi regular if and only if LR is quasi complemented and for every a E LR, there exists r E R such that a* = D*(r). This result will be proved after the following preliminary lemma.

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Section: Theorem 7 a Semiprime Ring R Is A Pf-ring If And Only If Lrmentioning

confidence: 84%

“…Hence D : ~R ~ ~LR is a lattice isomorphism. So, by Theorems 1, 5, and 6, we get the following theorem that answers a question raised by Simmons [9].…”

Section: Theorem 6 a Lattice L Is Conormal If And Only If Each Minimmentioning

confidence: 87%

“…In this paper we continue this theme and answer a couple of questions raised by Simmons [9]. Then we proceed to prove further results in that direction.…”

Section: D(1r)=i D(0r)=0 D(ab) = D(a) a D(b) D(a + B) <= D(a) V Dmentioning

confidence: 96%

Section: D(1r)=i D(0r)=0 D(ab) = D(a) a D(b) D(a + B) <= D(a) V Dmentioning

confidence: 99%

mentioning

confidence: 99%

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Further results on reticulated rings

Al-Ezeh

1992

Acta Math Hung

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On the Maximality of Some Conormal Extensions of a Lattice

Gehrke

Kaiser

1987

Mathematical Logic Qtrly

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Let f : D --* E be an embedding between bounded distributive lattices. We call f normal if for every prime filter p of D the filter base f ( p ) is contained in a unique maximal filter of E. Dually, f will be called conormul if f ( p ) generates a prime filter in E. A lattice D is normal [4] if every prime filter p of D is contained in a unique maximal filter, i.e., if the identity on D is normal. Lattices with the dual property, i.e., every prime filter p of D contains a unique minimal filter, have been called conormal [lo]. If B is the Boolean center of D , then D is called complementedly normal if the inclusion B 4 D is normal. The lattice U of first order universal sentences provides an important example of a complementedly normal lattice, being a normal extension of its boolean center B of all open sentences (cf. [S]); similarly, the lattice E of existential sentences is a conormal extension of B. It is easy to see that for any extension f : D + E a conormal intermediate extension E' 2 D is contained i n a maximal conormal extension E E E . Now, given any conormal extension f : D -+ E it is natural to ask whether it is maximal with respect to certain extensions of E . Our paper deals with two basic cases. Firstly, we look a t the booleanization E* of the lattice E . We give a necessary and sufficient condition for the conormal extension f : D -+ E to be maximal within D + E -+ E*. Our theorem applies t o the Stone representation of D as the lattice of compact open sets within all open sets of the Stone space for D, as well as for the aforementioned example of the lattice of existential sentences. I n recent years logicians have become interested in sober spaces (cf. [5] for the connection with complete Heyting algebras). For sober spaces (which have to be To) a separation axiom which lies between To and T I , the so called T,-axiom, is quite useful. We shall give a characterization of those Stone spaces which satisfy the T,-axiom in terms of conormality: The lattice of open sets of S(D) is a maximal conormal extension within the power set of S(D). This T,-axiom also has a strong impact on the order structure of S(D), namely, any chain of prime filters has a maximum. 1. , 411 lattices in this paper are bounded and distributive. Homomorphisms preserve 0 and 1, and embeddings are fAjective homomorphisms. The Stone space S(D) of a lattice D consists of the prime filters p of D together with the compact-open sets V ( T ) = { p I x E p } as a base for the lattice O(S(D)) of all open sefs. If f : D -+*E is a conormal extension then it is easy to see that the Stone map Q = S ( f ) : X(E) +;X(D), q --* f-'(q), has a left adjoint 6 : S(D) 4 X(E), p F+ {y 1 y 2 f ( x ) , x E p } , i.e., 6 is continuous and monotone and one has o(p) s q if and only if p s e(q). Then for an element y of E , the counter image of the compact-open set V(y) of X(E) is open in S(D).This yields a homomorphism g : E + U(X(D)), y t , c~-~(V(y)). As in [9] we say that

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