A Bazzoni-Type Theorem for Multiplicative Lattices

Abstract

We prove a Bazzoni-type theorem for multiplicative lattices thus unifying several ring/monoid theoretic results of this type.

1 Introduction

Let D be an integral domain. Consider the following two assertions:

(i) If I is an ideal of D whose localizations at the maximal ideals are finitely generated, then I is finitely generated.

(ii) Every $x\in D-\{0\}$ belongs to only finitely many maximal ideals of D.

While $(ii)\Rightarrow (i)$ is well-known and easy to prove, Bazzoni [3, p. 630] conjectured that the converse is true for Prüfer domains. Recall that D is a Prüfer domain if every finitely generated ideal I of D is locally principal.

Holland et al. [10, Theorem 10] proved Bazzoni’s conjecture for Prüfer domains using techniques from lattice-ordered groups theory and McGovern [14, Theorem 11] proved the same result using a direct ring theoretic approach. Halter-Koch [9, Theorem 6.11] proved Bazzoni’s conjecture for r-Prüfer monoids (see Section 4). Zafrullah [16, Proposition 5] proved Bazzoni’s conjecture for Prüfer v-multiplication domains. Finocchiaro and Tartarone [6, Theorem 4.5] proved Bazzoni’s conjecture for almost Prüfer ring extensions (see Section 3). Recently, Chang and Hamdi [4, Theorem 2.4] proved Bazzoni’s conjecture for almost Prüfer v-multiplication domains (see Section 4).

The purpose of this paper is to prove a Bazzoni-type theorem for multiplicative lattices (see Section 2), thus unifying the results mentioned above (see Sections 3 and 4). Our standard references are [1, 7, 8].

2 Main Result

We use an abstract ideal theory approach, so we work with multiplicative lattices.

Definition 1

A multiplicative lattice is a complete lattice $(L,\le )$ (with bottom element 0 and top element 1) which is also a multiplicative commutative monoid with identity 1 (the top element) and satisfies $a( \bigvee b_\alpha ) = \bigvee ab_\alpha$ for each $a,b_\alpha \in L$.

Let L be a multiplicative lattice. The elements in $L-\{1\}$ are said to be proper. Denote by Max(L) the set of maximal elements of L. For $x,y\in L$, set $(y : x) =\bigvee \{a \in L ;\ ax \le y\}$.

We recall some standard terminology.

Definition 2

Let L be a multiplicative lattice and let $x,p \in L$.

(1) p is prime if $p\ne 1$ and for all $a,b \in L$, $ab \le p$ implies $a \le p$ or $b \le p$. It follows easily that every maximal element is prime.

(2) x is compact if whenever $x\le \bigvee _{y\in S} y$ with $S\subseteq L$, we have $x\le \bigvee _{y\in T} y$ for some finite subset T of S.

(3) L is a C -lattice if the set $L^*$ of compact elements of L is closed under multiplication, $1\in L^*$ and every element in L is a join of compact elements.

(4) x is meet-principal if $y \wedge zx = ((y : x) \wedge z) x$ for all $y, z \in L$ (in particular $(y:x)x=x\wedge y$).

(5) x is join-principal if $y \vee (z : x) = ((yx \vee z) : x)$ for all $y, z \in L$ (in particular $(xy:x)=y\vee (0:x)$).

(6) x is cancellative if for all $y,z\in L$, $xy = xz$ implies $y = z$.

(7) x is CMP (ad hoc name) if x is cancellative and meet-principal.

(8) L is a lattice domain if $(0:a)=0$ for all $a\in L-\{0\}$.

In the sequel, we work with C-lattices and their localization theory. Let L be a C-lattice. For $p \in L$ a prime element and $x\in L$, we set

$$x_p = \bigvee \{a \in L^* ;\ ab \le x \text{ for } \text{ some } b \in L^* \text{ with } b\not \le p\}.$$

Then $L_p:=\{ x_p;\ x\in L\}$ is again a lattice with multiplication $(x_p,y_p)\mapsto (xy)_p=(x_py_p)_p$, join $\{ (b_\alpha )_p \}\mapsto (\bigvee (b_\alpha )_p)_p = (\bigvee b_\alpha )_p$ and meet $\{ (b_\alpha )_p \}\mapsto (\bigwedge (b_\alpha )_p)_p$. The next lemma collects several basic properties.

Lemma 3

Let L be a C-lattice, let $x,y\in L$ and let $p \in L$ be a prime element.

(1) $x_p = 1$ if and only if $x\not \le p$.

(2) $(x \wedge y)_p = x_p \wedge y_p$.

(3) If x is compact, then $(y : x)_p = (y_p : x_p)$.

(4) $x=y$ if and only if $x_m=y_m$ for each $m\in Max(L)$.

(5) A cancellative element x is CMP if and only if $(y:x)x=x\wedge y$ for all $y\in L$.

(6) If x is compact, then $x_p$ is compact in $L_p$. Conversely, if $x_p$ is compact in $L_p$, then $x_p=c_p$ for some compact element $c\le x$.

(7) If x and y are CMP elements, then so is xy.

(8) If x is compact, y is CMP and $x\le y$, then (x : y) is compact.

Proof

For (1–4) see [11, pp. 201–202], for (5) see [15, Lemma 2.10], while (6–7) follow easily from definitions. We prove (8). Note that $(x:y)y=x\wedge y = x$. Suppose that $(x:y)\le \bigvee _{i\in A} z_{i}$. Then $x=(x:y)y\le \bigvee _{i\in A} z_{i}y$, so $(x:y)y\le \bigvee _{i\in B} z_{i}y$ for some finite subset B of A. Cancel y to get $(x:y)\le \bigvee _{i\in B} z_{i}$. $\square$

Say that x and $y \in L$ are comaximal if $x\vee y=1$. Clearly, $x\vee yz=1$ if and only if $x\vee y=1$ and $x\vee z=1$. When $t\le u$ we say that t is below u or that u is above t.

Lemma 4

Let L be a C-lattice and $z\in L-\{1\}$ a compact element such that $\{ m\in Max(L);\ z\le m\}$ is infinite. There exists an infinite set $\{ a_n;\ n\ge 1\}$ of pairwise comaximal proper compact elements such that $z\le a_n$ for each n.

Proof

We may clearly assume that $z=0$ (just change L by $\{ x\in L;\ x\ge z\}$). Say that a proper compact element h is big (ad hoc name) if h is below only one maximal element M(h). We separate in two cases.

Case (1): Every proper compact element is below some big compact element. We proceed by induction. Suppose that $n\ge 1$ and we already have big compacts $a_1$,...,$a_n$ such that $M(a_1)$,...,$M(a_n)$ are distinct maximal elements (for $n=1$ just pick an arbitrary big compact $a_1$). Let p be a maximal element other than $M(a_1)$,...,$M(a_n)$. There exists a compact element $c\le p$ such that $c\not \le M(a_i)$ for $1\le i\le n$ (take $c=c_1\vee \cdots \vee c_n$ where each $c_i\in L^*$ satisfies $c_i\le p$ and $c_i\not \le M(a_i)$). Then take a big compact element $a_{n+1}\ge c$. This way we construct an infinite set $\{ a_n;\ n\ge 1\}$ of big compacts such that all $M(a_n)$’s are distinct. Hence the $a_n$’s are pairwise comaximal.

Case (2): There exists a proper compact element $a_0$ which is not below any big compact element. Clearly every proper compact above $a_0$ inherits this property. Pick two distinct maximal elements p and q above $a_0$. As L is a C-lattice, there exist two comaximal compacts $a_1\le p$ and $b_1\le q$ (note that $p\vee q=1$, express p and q as joins of compact elements and use the fact that 1 is compact). Repeating this argument for $a_1$, there exist two comaximal proper compacts $a_2\ge a_1$ and $b_2\ge a_1$. Note that $b_2$ and $b_1$ are comaximal. Thus we construct inductively an infinite set $\{b_n ;\ n\ge 1\}$ of pairwise comaximal proper compact elements. $\square$

In a C-lattice L, we say that an element x is locally compact if $x_m$ is compact in $L_m$ for each $m\in Max(L)$. We state our main result which is a Bazzoni-type theorem for C-lattices.

Theorem 5

Let L be a C-lattice domain satisfying the following two conditions:

(a) every nonzero element is above some cancellative compact element, and

(b) every compact element $x\ne 1$ has some power $x^n$ below some proper CMP element.

Then the following conditions are equivalent:

(i) Every locally compact element of L is compact.

(ii) Every nonzero element is below at most finitely many maximal elements.

Proof

$(ii) \Rightarrow (i).$ Although this part is well-known and easy, we include a proof for the reader’s convenience. Let x be a nonzero locally compact element of L and let $a\le x$ be a nonzero compact element. By (ii), there are only finitely many maximal elements above a, say $m_1$,...,$m_k$. For each i between 1 and k, pick a compact element $c_i\le x$ such that $x_{m_i}=({c_i})_{m_i}$. A local check shows that $x=a\vee c_1 \vee \cdots \vee c_k$, so x is compact. Note that this part works for any C-lattice.

$(i) \Rightarrow (ii)$. Deny, so suppose that some nonzero element c is below infinitely many maximal elements. By hypothesis (a), we may assume that c is a cancellative compact element. By Lemma 4 and hypothesis (b), there exist an infinite set $\{ b_n;\ n\ge 1\}$ of proper pairwise comaximal CMP elements and integers $k_n\ge 1$ such that $c^{k_n}\le b_n$ for $n\ge 1$ ($k_n$ minimal with this property). Restricting to a subsequence, we may assume that $k_n\le k_{n+1}$ for all n. We then have $c^{k_n}\le b_1\wedge \cdots \wedge b_n=b_1\cdots b_n$ for all n.

Claim $(*):$ The element $a:=\bigvee _{n\ge 1} (c^{k_n}:b_1\cdots b_n)$ is locally compact.

Pick $m\in$ Max(L). Since the $b_n$’s are pairwise comaximal, m is above at most one of them. Assume first that $m\ge b_s$. Since each product $b_1\cdots b_n$ is compact, we get

$$a_m=(\bigvee _{n\ge 1} ((c^{k_n})_m:(b_1\cdots b_n)_m))_m= (c^{k_1}\vee (c^{k_s}:b_{s}))_m$$

which is compact in $L_m$, cf. Lemma 3. Similarly, when m is above no $b_n$, we get $a_m=(c^{k_1})_m$, so $a_m$ is compact in $L_m$, hence Claim $(*)$ is proved. By (i), a is compact. So $a=\bigvee _{n= 1}^q (c^{k_n}:b_1\cdots b_n)$ for some $q\ge 1$. We get

$$ (c^{k_{q+1}}:b_1\cdots b_{q+1}) \le (c^{k_1}:b_1\cdots b_{q}) $$

so multiplying by $b_1\cdots b_{q+1}$ (which is a CMP element) and taking into account that $c^{k_{q+1}}\le b_1\cdots b_{q+1}$, we get

$$ c^{k_{q+1}} \le (c^{k_1}:b_1\cdots b_{q})b_1\cdots b_{q+1}\le c^{k_1}b_{q+1}. $$

Since ${k_{q+1}} \ge k_1$ and $c^{k_1}$ is cancellative, we get $c^{k_{q+1}-k_1}\le b_{q+1}$, which is a contradiction since $k_{q+1}$ was minimal with $c^{k_{q+1}}\le b_{q+1}$. $\square$

Recall that a C-lattice domain is a Prüfer lattice if every compact element is principal (i.e., meet-principal and join-principal). In a C-lattice domain, every nonzero join-principal element x is cancellative (because $(yx:x)=y\vee (0:x)=y$ for each y). So in a Prüfer lattice domain every nonzero compact element is CMP.

Corollary 6

Let L be a C-lattice domain in which every nonzero compact element is CMP (e.g., a Prüfer lattice domain). Then conditions (i) and (ii) of Theorem 5 are equivalent.

Bazzoni’s conjecture for Prüfer domains [10, Theorem 10] (see Introduction) follows from Corollary 6 since the ideal lattice of a Prüfer domain is clearly a Prüfer lattice.

3 Almost Prüfer Extensions

We recall several definitions from [6, 12]. Let $A\subseteq B$ be a commutative ring extension and I an ideal of A. Then I is called B -regular if $IB=B$ and I is called B -invertible if $IJ=A$ for some A-submodule J of B. Every B-invertible ideal is B-regular, since $A=IJ\subseteq IB$ implies $IB=B$. We say that $A \subseteq B$ is an almost Prüfer extension if every finitely generated B-regular ideal of A is B-invertible.

Finocchiaro and Tartarone [6, Theorem 4.5] proved Bazzoni’s conjecture for almost Prüfer ring extensions. We state their result and derive it from Corollary 6.

Theorem 7

(Finocchiaro and Tartarone) If $A\subseteq B$ is an almost Prüfer extension, the following are equivalent:

(i) Every B-regular locally principal ideal of A is B-invertible.

(ii) Every B-regular ideal of A is contained in only finitely many maximal ideals of A.

Proof

It is well-known and easy to prove that (ii) implies (i), see [6, Corollary 3.5]. We prove the converse. Let L be the set of all B-regular ideals of A together with the zero ideal and order L by inclusion. As shown in [15, Lemma 7.1], L is a C-lattice domain under usual ideal multiplication, where the join is the ideal sum and the meet is the ideal intersection except the case when we get a non-B-regular ideal when we put $\bigwedge =0$. By [15, Lemma 7.1], the set $L^*$ of compact elements in L is exactly the set of (B-regular) finitely generated ideals of A together with the zero ideal. After this preparation it becomes clear that $[(i)\Rightarrow (ii)]$ follows from Corollary 6 provided we prove the two claims below. Write $x\in L$ as $\widehat{x}$ when considered as an ideal of A.

Claim 1: Every nonzero compact element of L is a CMP element.

Let c be a nonzero compact element of L. As $A\subseteq B$ is almost Prüfer, $\widehat{c}$ is a B-invertible ideal, so $\widehat{c}J=A$ for some A-submodule J of B. Then c is clearly cancellative. By Lemma 3, it suffices to show that $(x:c)c=x\wedge c$ for each $x\in L$. Changing x by $x\wedge c$, we may assume that $x\le c$. We have $\widehat{x}=\widehat{x}J\widehat{c}$, so $x=yc$ where $y\in L$ is such that $\widehat{y}=\widehat{x}J$ (note that $\widehat{x}J\subseteq A$). From $x=yc$ we get $y\le (x:c)$, so $x=yc\le (x:c)c\le x$, thus $(x:c)c=x$.

Claim 2: Every locally compact element of L is compact.

Suppose that c is a nonzero locally compact element of L. Let m be a maximal element of L, that is, $\widehat{m}$ is a B-regular maximal ideal of A. So $c_m=\bigvee \{y\in L^*;\ ys\le c$ for some $s\in L^*,s\not \le m\}$ is compact in the lattice $L_m=\{x_m;\ x\in L\}$. Then $c_m=h_m$ for some $h\in L^*$. Extending these ideals in $A_{\widehat{m}}$, we get $\widehat{c}A_{\widehat{m}} = \widehat{c_m}A_{\widehat{m}}= \widehat{h_m}A_{\widehat{m}} = \widehat{h}A_{\widehat{m}}$. Since $A\subseteq B$ is almost Prüfer, $\widehat{h}$ is B-invertible. By [12, Proposition 2.3], $\widehat{h}A_{\widehat{m}} = \widehat{c}A_{\widehat{m}}$ is a principal ideal of $A_{\widehat{m}}$. Thus $\widehat{c}$ is a locally principal ideal of A. By (i), $\widehat{c}$ is B-invertible, so $\widehat{c}$ is finitely generated, cf. [12, Proposition 2.3]. Thus c is compact in L. $\square$

4 Ideal Systems on Monoids and Integral Domains

Let H be a commutative multiplicative monoid (with zero element 0 and unit element 1) such that every nonzero element of H is cancellative and let $\mathcal {P}(H)$ be the power set of H. A map $r:\mathcal {P}(H)\rightarrow \mathcal {P}(H)$, $X\mapsto X_r$, is an ideal system on H if the following conditions hold for all $X,Y\in \mathcal {P}(H)$ and $c\in H$:

$$(1)\ cX_r = (cX)_r, (2)\ X\subseteq X_r, (3)\ X\subseteq Y \text{ implies } X_r\subseteq Y_r, (4)\ (X_r)_r=X_r.$$

Then $X_r$ is called the r -closure of X and a set of the form $X_r$ is called an r-ideal. An r-ideal I is r -finite if $I=Y_r$ for some finite subset Y of I. The ideal system r is called finitary if $X_r=\bigcup \{ Y_r;\ Y\subseteq X$ finite$\}$.

Assume that r is finitary. A proper r-ideal P is prime if, for $x,y\in H$, $xy\in P$ implies $x\in P$ or $y\in P$. The localization of H at P is the fraction monoid $H_P=\{ x/t;\ x\in H, t\in H-P\}$ which comes together with the finitary ideal system $r_P$ defined by $\{a_1/s_1,...,a_n/s_n\}_{r_P}=(\{a_1,...,a_n\}_r)_P$ for all $a_1,...,a_n\in H$ and $s_1,...,s_n\in H-P$. A maximal r -ideal is a maximal element of the set of proper r-ideals of H. Any proper r-ideal is contained in a maximal one and a maximal r-ideal is prime. A nonzero r-ideal I is r -invertible if I is r-finite and r -locally principal (i.e., $I_M=\{ x/t;\ x\in I, t\in H-M\}=yH_M$ with $y\in I_M$ (depending on M) for all maximal r-ideals M). Next H is an r -Prüfer monoid if every nonzero r-finite r-ideal of H is r-invertible. For complete details we refer to [8].

Halter-Koch [9, Theorem 6.11] proved Bazzoni’s conjecture for r-Prüfer monoids. We state his result and derive it from Corollary 6.

Theorem 8

(Halter-Koch) If H is an r-Prüfer monoid for some finitary ideal system r on H, the following are equivalent.

(i) Every r-locally principal r-ideal of H is r-finite.

(ii) Every nonzero r-ideal of H is contained in only finitely many maximal r-ideals.

Proof

It is well-known and easy to prove that (ii) implies (i). We prove the converse. Let L be the set of all r-ideals of H ordered by inclusion. As shown in [8, Chapter 8], L is a C-lattice domain under r-ideal multiplication $(I,J)\mapsto (IJ)_r$, join $\bigvee \{ J_\alpha \}:=(\bigcup J_\alpha )_r$ and meet $\bigwedge \{ J_\alpha \}:=\bigcap J_\alpha$. By [15, Lemma 8.1], the set $L^*$ of compact elements in L is exactly the set of r-finite r-ideals of H. After this preparation it becomes clear that $[(i)\Rightarrow (ii)]$ follows from Corollary 6 provided we prove the two claims below. Write $x\in L$ as $\widehat{x}$ when considered as an r-ideal of H.

Claim 1: Every nonzero compact element of L is a CMP element.

Let $c\in L$ be a nonzero compact, in other words $\widehat{c}$ is nonzero r-finite r-ideal. Since H is r-Prüfer, $\widehat{c}$ is r-invertible. So c is a CMP element of L, cf. [15, Lemma 8.2].

Claim 2: Every locally compact element of L is compact.

Suppose that c is a locally compact element of L. Let m be a maximal element of L, that is, $\widehat{m}$ is a maximal r-ideal of H. So $c_m=\bigvee \{y\in L^*;\ ys\le c$ for some $s\in L^*,s\not \le m\}$ is compact in the lattice $L_m=\{x_m;\ x\in L\}$. Then $c_m=h_m$ for some $h\in L^*$. Switching to H, we have $(\widehat{c})_{\widehat{m}}=(\widehat{c_m})_{\widehat{m}}= (\widehat{h_m})_{\widehat{m}}=\widehat{h}_{\widehat{m}}$. Since H is an r-Prüfer monoid, $\widehat{h}_{\widehat{m}}$ is a principal $r_{\widehat{m}}$-ideal of $H_{\widehat{m}}$, cf. [8, Theorem 12.3]. Thus $\widehat{c}$ is an r-locally principal r-ideal. By (i), $\widehat{c}$ is r-finite, thus c is compact in L. $\square$

Next we present an application for integral domains. Let D be an integral domain. The t ideal system on D is defined by $X_t=\bigcup \{ Y_v;\ Y\subseteq X \text{ finite }\}$ for all $X\subseteq D$, where $Y_v=\bigcap \{ (aD:_D b);\ a,b\in D, bY\subseteq aD\}.$ Clearly t is finitary, so the set $Max_t(D)$ of maximal t-ideals is nonempty, see [8, Chapter 11].

The w ideal system on D is defined by $X_w=\bigcap \{ XD_M;\ M\in Max_t(D)\}$ for all $X\subseteq D$, where $XD_M$ is the ideal generated by X in $D_M$. So a w-ideal is an ideal of the ring D. For $X\subseteq D$, we have $X_w=(XD)_w$ and $X_wD_M=XD_M$ for each $M\in Max_t(D)$. Moreover, w is finitary and the set of maximal w-ideals is exactly $Max_t(D)$. A w-finite ideal has the form $((a_1,...,a_n)D)_w$ for some $a_i$’s in D. And a nonzero w-ideal is w -invertible if it is w-finite and t-locally principal (i.e., $ID_M$ is a principal ideal of $D_M$ for each $M\in Max_t(D))$. For details on the w ideal system we refer to [2, 5].

According to [13], D is an almost Prüfer v -multiplication domain (in short APVMD) if for every $a_1,...,a_n\in D-\{0\}$, the ideal $((a^k_1,...,a^k_n)D)_w$ is w-invertible for some $k\ge 1$. Say that a w-ideal I of D is t -locally finitely generated, if $ID_M$ is a finitely generated ideal of $D_M$ for each $M\in Max_t(D)$.

Chang and Hamdi [4, Theorem 2.4] proved Bazzoni’s conjecture for APVMDs. We state their result and derive it from Theorem 5.

Theorem 9

(Chang and Hamdi) For an APVMD D, the following statements are equivalent:

(i) Each nonzero t-locally finitely generated w-ideal of D is w-finite.

(ii) Every nonzero ideal of D is contained in only finitely many maximal t-ideals.

Proof

It is well-known and easy to prove that (ii) implies (i), see for instance the proof of $(3)\Rightarrow (1)$ in [4, Theorem 2.4]. We prove that (i) implies (ii). Let L be the set of all w-ideals of D ordered by inclusion. As shown in [8, Chapter 8], L is a C-lattice domain under w-ideal multiplication $(I,J)\mapsto (IJ)_w$, join $\bigvee \{ J_\alpha \}:=(\bigcup J_\alpha )_w$ and meet $\bigwedge \{ J_\alpha \}:=\bigcap J_\alpha$. By [15, Lemma 8.1], the set $L^*$ of compact elements in L is exactly the set of w-finite ideals of D. Condition (a) of Theorem 5 holds clearly for L (any nonzero ideal contains a nonzero principal ideal). After this preparation it becomes clear that $[(i)\Rightarrow (ii)]$ follows from Theorem 5 provided we prove the two claims below. Write $x\in L$ as $\widehat{x}$ when considered as a w-ideal of D.

Claim 1: Every $d\in L^*-\{1\}$ has some power $d^n$ below some proper CMP element.

Let $d\in L^*-\{1\}$. Then $\widehat{d}=(a_1,...,a_k)_w$ for some elements $a_i\in \widehat{d}$. Since D is an APVMD, $(a^s_1,...,a^s_k)_w$ is w-invertible for some $s\ge 1$. If $(a^s_1,...,a^s_k)_w=\widehat{f}$ with $f\in L$, then f is a proper CMP element of L, cf. [15, Lemma 8.2]. Moreover $d^{sk}\le f$, so Claim 1 is proved.

Claim 2: Every locally compact element of L is compact.

Suppose that c is a locally compact element of L. Let m be a maximal element of L, that is, $\widehat{m}$ is a maximal w-ideal of D. So $c_m=\bigvee \{y\in L^*;\ ys\le c$ for some $s\in L^*,s\not \le m\}$ is compact in the lattice $L_m=\{x_m;\ x\in L\}$. Then $c_m=h_m$ for some $h\in L^*$. Hence $\widehat{h}=(b_1,...,b_n)_w$ for some elements $b_i\in \widehat{h}$. Switching to D, we have

$$\widehat{c}D_{\widehat{m}}=\widehat{c_m}D_{\widehat{m}}= (\widehat{h_m})D_{\widehat{m}}=\widehat{h}D_{\widehat{m}} =(b_1,...,b_n)_wD_{\widehat{m}}=(b_1,...,b_n)D_{\widehat{m}}.$$

Thus $\widehat{c}$ is a t-locally finitely generated ideal of D. By (i), $\widehat{c}$ is w-finite, thus c is compact in L. $\square$