Classification of multiplication modules over multiplication rings with finitely many minimal primes

Abstract

A classification of multiplication modules over multiplication rings with finitely many minimal primes is obtained. A characterization of multiplication rings with finitely many minimal primes is given via faithful, Noetherian, distributive modules. It is proven that for a multiplication ring with finitely many minimal primes every faithful, Noetherian, distributive module is a faithful multiplication module, and vice versa.

1. Introduction

In this paper, all rings are commutative with 1 and all modules are unital. A ring $R$ is called a multiplication ring if $I$ and $J$ are ideals of $R$ such that $J\subseteq I$ then $J=I^{\prime }I$ for some ideal $I^{\prime }$ of $R$ . An $R$ -module $M$ is called a multiplication moduleif each submodule of $M$ is equal to $IM$ for some ideal $I$ of the ring $R$ . The concept of multiplication ring was introduced by Krull in [5]. In [6], Mott proved that a multiplication ring has finitely many minimal prime ideals iff it is a Noetherian ring.

The next theorem is a description of multiplication rings with finitely many minimal primes.

Theorem 1.1. ([1 , Theorem 1.1]) Let $R$ be a ring with finitely many minimal prime ideals. Then the ring $R$ is a multiplication ring iff $R\cong \displaystyle \prod _{i=1}^{n} D_{i}$ is a finite direct product of rings where $D_{i}$ is either a Dedekind domain or an Artinian, local principal ideal ring.

Classification of multiplication modules over multiplication rings with finitely many minimal primes. Using Theorem 1.1, a criterion for a direct sum of modules to be a multiplication module (Theorem 2.1) and some other results, a classification of multiplication modules over a multiplication ring with finitely many minimal primes is given, Theorem 1.2.

Theorem 1.2. Let $R$ be a multiplication ring with finitely many minimal primes, that is $R\cong \displaystyle \prod _{i=1}^{n} D_{i}$ is a finite direct product of rings where $D_{i}$ is either a Dedekind domain or an Artinian, local principal ideal ring and $1=e_1+\cdots + e_n$ be the corresponding sum of orthogonal idempotents of the ring $R$ . Let $M$ be an $R$ -modules and $M=\oplus _{i=1}^n M_i$ where $M_i:= e_iM$ . Then the $R$ -module $M$ is a multiplication $R$ -module iff each $D_i$ -module $M_i$ is either isomorphic to $D_i$ or to $D_i/I_i$ where $I_i$ is a nonzero ideal of $D_i$ or to a nonzero ideal of the ring $D_i$ in case when the ring $D_i$ is a Dedekind domain.

Classification of faithful multiplication modules over a multiplication ring with finitely many minimal primes.

Theorem 1.3. Let $R$ be a multiplication ring with finitely many minimal primes. We keep the notation of Theorem 1.2 ( $R\cong \displaystyle \prod _{i=1}^{n} D_{i}$ ). Then an $R$ -module $M=\oplus _{i=1}^n M_i$ (where $M_i= e_iM$ ) is a faithful multiplication $R$ -module iff for each $i=1, \ldots, n$ , either ${}_RM_i\simeq D_i$ or ${}_RM_i\simeq I_i$ where $I_i$ is a nonzero ideal of the ring $D_i$ in case when $D_i$ is a Dedekind domain.

Proof. The theorem follows at once from Theorem 1.2.

Characterization of multiplication rings with finitely many minimal primes via faithful, Noetherian, distributive modules. Let $R$ be a ring and $M$ be an $R$ -module. A submodule $N$ of $M$ is called a distributive submodule if one of the following equivalent conditions holds: For any submodules $M_1$ and $M_2$ of $M$ ,

\begin{eqnarray*} (M_1+M_2)\cap N &=& M_1\cap N +M_2\cap N, \\ M_1\cap M_2+ N &=&(M_1+N)\cap (M_2+N). \end{eqnarray*}

The $R$ -module $M$ is called a distributive module if all submodules of $M$ are distributive submodules.

Theorem 1.4. A commutative ring $R$ is a multiplication ring with finitely many minimal primes iff there is a faithful, Noetherian, distributive $R$ -module.

Classification of faithful, Noetherian, distributive modules over a multiplication ring with finitely many minimal primes.

Theorem 1.5. Let $R$ be a multiplication ring with finitely many minimal primes. Then every faithful, Noetherian, distributive $R$ -module is a faithful multiplication $R$ -module, and vice versa.

2. Proofs

In this section, we prove the results from the Introduction.

Definition 2.1. We say that the intersection condition holds for a direct sum $M=\bigoplus _{\lambda \in \Lambda } M_{\lambda }$ of nonzero $R$ -modules $M_{\lambda }$ if for all submodules $N$ of $M$ , $N= \bigoplus _{\lambda \in \Lambda } (N\bigcap M_\lambda )$ .

Definition 2.2. Let $M=\bigoplus _{\lambda \in \Lambda } M_{\lambda }$ be a direct sum of nonzero $R$ -modules with $\textrm{card}(\Lambda )\geqslant 2$ , $\mathfrak{a}_{\lambda }=\textrm{ann}_{R}(M_{\lambda })$ and $\mathfrak{a}^{\prime }_{\lambda }=\cap _{\mu \neq \lambda }\mathfrak{a}_{\mu }$ . We say that the orthogonality condition holds for the direct sum $M=\bigoplus _{\lambda \in \Lambda } M_{\lambda }$ if $\mathfrak{a}^{\prime }_{\lambda }M_{\mu }= \delta _{\lambda \mu }M_{\mu }$ for all $\lambda, \mu \in \Lambda$ . Clearly, $\mathfrak{a}^{\prime }_{\lambda }\neq 0$ for all $\lambda \in \Lambda$ (since all $M_{\lambda } \neq 0$ ). In particular, $\mathfrak{a}_{\lambda } \neq 0$ for all $\lambda \in \Lambda$ .

Definition 2.3. Let $M=\bigoplus _{\lambda \in \Lambda } M_{\lambda }$ be a direct sum of nonzero $R$ -modules with $\textrm{card}(\Lambda ) \geq 2$ . We say that the strong orthogonality condition holds for $M$ if for each set of $R$ -modules $\lbrace N_{\lambda } \rbrace _{\lambda \in \Lambda }$ such that $N_{\lambda }\subseteq M_{\lambda }$ , there is a set of ideals $\lbrace I_{\lambda } \rbrace _{\lambda \in \Lambda }$ of $R$ such that $I_{\lambda } M_{\mu }=\delta _{\lambda \mu }N_{\lambda }$ for all $\lambda, $ $\mu \in \Lambda$ where $\delta _{\lambda \mu }$ is the Kronecker delta. The set of ideals $\lbrace I_{\lambda } \rbrace _{\lambda \in \Lambda }$ is called an orthogonalizer of $\lbrace N_{\lambda } \rbrace _{\lambda \in \Lambda }$ .

Theorem 2.1 is one of the criteria for a direct sum of modules to be a multiplication module that are obtained in [1]. It is given via the intersection and strong orthogonality conditions.

Theorem 2.4. ([2]) Let $M=\bigoplus _{\lambda \in \Lambda } M_{\lambda }$ be a direct sum of nonzero $R$ -modules with $\textrm{card}(\Lambda ) \geq 2$ . Then $M$ is a multiplication module iff the intersection and strong orthogonality conditions hold for the direct sum $M=\bigoplus _{\lambda \in \Lambda } M_{\lambda }$ .

An $R$ -module is called a cyclic if it is 1-generated. For an $R$ -module $M$ , let $\textrm{Cyc}_R(M)$ be the set of its cyclic submodules. For an $R$ -module $M$ , we denote by $\textrm{ann}_{R}(M)$ its annihilator. An $R$ -module $M$ is called faithful if $\textrm{ann}_{R}(M)=0$ . For a submodule $N$ of $M$ , the set $[N:M]:= \textrm{ann}_R(M/N)= \{ r\in R\, | \, rM \subseteq N\}$ is an ideal of the ring $R$ that contains the annihilator $\textrm{ann}_R(M)=[0:M]$ of the module $M$ . The set ${\theta }(M):=\sum _{C\in \textrm{Cyc}_R(M)}[C:M]$ is an ideal of $R$ . Clearly, $\textrm{ann}_R(M)\subseteq{\theta }(M)$ . If $M$ is an ideal of $R$ then $M\subseteq \theta (M)$ .

Proof of Theorem 1.2. $(\Leftarrow )$ All the $D_i$ -modules $M_i$ of the theorem are multiplication $D_i$ -modules. Hence, the direct sum $\oplus _{i=1}^nM_i$ is a multiplication module over the direct product rings $R=\prod _{i=1}^nD_i$ .

$(\Rightarrow )$ Suppose that the $R$ -module $M=\oplus _{i=1}^n M_i$ is a multiplication $R$ -module where $M_i=e_iM$ for $i=1, \ldots, n$ . We have the following claims.

(i) The $D_i$ -module $M_i$ is a multiplication $D_i$ -module: The statement is obvious since $R=\prod _{i=1}^nD_i$ .

(ii) The $D_i$ -module $M_i$ is a finitely generated $D_i$ -module: Since $M_i$ is a multiplication $D_i$ -module,

\begin{equation*}M_i=\sum _{C\in \textrm {Cyc}_{D_i}(M_i)}C=\sum _{C\in \textrm {Cyc}_{D_i}(M_i)}[C:M_i]M_i=(\sum _{C\in \textrm {Cyc}_{D_i}(M_i)}[C:M_i])M_i={\theta } (M_i)M_i.\end{equation*}

The ideal ${\theta } (M_i)=\sum _{C\in \textrm{Cyc}_{D_i}(M_i)}[C:M_i]$ of the Noetherian ring $D_i$ is a finitely generated $D_i$ -module, that is, ${\theta } (M_i) = \sum _{i=1}^{n_i}D_i{\theta }_i$ for some elements ${\theta }_i\in{\theta } (M_i)$ . Then

\begin{equation*}M_i={\theta } (M_i) M_i=\sum _{i=1}^{n_i}D_i{\theta }_iM_i\subseteq \sum _{i=1}^{n_i}C_i\subseteq M_i,\end{equation*}

and so the $D_i$ -module $M_i=\sum _{i=1}^{n_i} C_i$ is finitely generated.

(iii) Suppose that the ring $D_i$ is a Dedekind domain. Then the $D_i$ -module $M_i$ is isomorphic either to $D_i$ or to $D_i/I_i$ or to $J_i$ where $I_i$ and $J_i$ are ideals of the ring $D_i$ : It is well-known that a nonzero finitely generated module $\mathcal{M}$ over a Dedekind domain $D$ is a direct sum $\mathcal{M} = \mathcal{F} \oplus\mathcal{T}$ of a torsion-free $D$ -module $\mathcal{F}$ and a torsion $D$ -module $\mathcal{T}$ ; $\mathcal{F} = I\oplus D^m$ for some ideal $I$ of $D$ and $m\geq 0$ ; and $\mathcal{T} =\oplus _{i=1}^{t_i}D/\mathfrak{p}_i^{m_i}$ where $\mathfrak{p}_i$ are maximal ideals of the ring $D$ and $m_i\in{\mathbb{N}}$ . Suppose that the $D$ -module $\mathcal{M}$ is a multiplication $D$ -module. By Theorem 2.1, the direct sum of $D$ -modules

\begin{equation*}\mathcal{M} = I\oplus D^m \oplus \bigoplus _{i=1}^{t_i} D/\mathfrak {p}_i^{m_i}\end{equation*}

must satisfy the strong orthogonality conditions. Hence, either $\mathcal{M} = I$ of $\mathcal{M} = D$ or $\mathcal{M} = \oplus _{i=1}^{t_i} D/\mathfrak{p}_i^{m_i}$ where $\mathfrak{p}_1, \ldots, \mathfrak{p}_{t_i}$ are distinct maximal ideals of the ring $D$ , and so $\mathcal{M} =\oplus _{i=1}^{t_i} D/\mathfrak{p}_i^{m_i}\simeq D/\prod _{i=1}^{t_i}\mathfrak{p}_i^{m_i}$ .

(iv) Suppose that $D_i$ is an Artinian, local, principal ideal ring. Then the $D_i$ -module $M_i$ is isomorphic either to $D_i$ or to $D_i/I_i$ where $I_i$ is a nonzero ideal of $D_i$ : Let $D=D_i$ and $\mathfrak{m}$ be the maximal ideal of the local ring $D_i$ and $\mathfrak{m}^{\nu } \neq 0$ and $\mathfrak{m}^{\nu +1} =0$ for some natural number $\nu$ . Then

\begin{equation*}\{ D, \mathfrak {m}, \mathfrak {m}^2, \ldots, \mathfrak {m}^\nu, \mathfrak {m}^{\nu +1} =0\}\end{equation*}

is the set of all the ideals of the ring $D$ . The $D$ -module $M_i$ is a nonzero finitely generated multiplication $D$ -module. Hence, $\{ M_i, \mathfrak{m} M_i, \mathfrak{m}^2 M_i, \ldots, \mathfrak{m}^\mu M_i, \mathfrak{m}^{\mu +1} M_i =0\}$ is the set of all $D$ -submodules of $M_i$ for some natural number $\mu$ such that $\mu \leq \nu$ . In particular, the $D$ -module $M_i$ is a uniserial $D$ -module since

\begin{equation*}M_i\supset \mathfrak {m} M_i \supset \mathfrak {m}^2 M_i\supset \cdots \supset \mathfrak {m}^\mu M_i\supset \mathfrak {m}^{\mu +1} M_i =0.\end{equation*}

Since the $D$ -module $M_i$ is a uniserial, we have that

\begin{equation*}{\textrm {dim }}_{k_{\mathfrak {m}}}(M_i/\mathfrak {m} M_i)=1\end{equation*}

where $k_{\mathfrak{m}} :=D/\mathfrak{m}$ , and so $M_i= Dm_i+\mathfrak{m} M_i$ for some element $m_i\in M_i\backslash \mathfrak{m} M_i$ . By the Nakayama Lemma, $M_i=Dm_i$ , and the statement (iv) follows.

Corollary 2.5. Let $R$ be an Artinian multiplication ring. Then every multiplication $R$ -module is an epimorphic image of the $R$ -module $R$ .

Proof. The corollary follows at once from Theorem 1.2.

Corollary 2.6. Let $R$ be a multiplication ring with finitely many minimal primes and $M$ be a multiplication $R$ -module. Then

1. 1. The endomorphism ring ${ \textrm{End }}_R(M)$ is also a multiplication ring.

2. 2. ${ \textrm{End }}_R(M)\simeq R/\textrm{ann}_R(M)$ .

3. 3. The ${ \textrm{End }}_R(M)$ -module $M$ is a faithful multiplication ${ \textrm{End }}_R(M)$ -module.

Proof. The corollary follows at once from Theorem 1.2.

In the proof of Theorem 1.4, we will use the following results.

Theorem 2.7. Let $R$ be a commutative ring.

1. 1. ([3 , Corollary, p. 177]) Let $M$ be a Noetherian distributive $R$ -module. Then every submodule of $M$ which is locally nonzero at every maximal ideal of $R$ , is of the form $IM$ where $I$ is a unique product of maximal ideals of $R$ .

2. 2. ([3 , Lemma 2.(ii)]) A finitely generated $R$ -module $M$ is a multiplication module iff the $R_{\mathfrak{p}}$ -module $M_{\mathfrak{p}}$ is a multiplication module for all prime/maximal ideals $\mathfrak{p}$ of $R$ .

3. 3. ([4 , Theorem1.3.(ii)]) (Cancellation Law) If $M$ is a finitely generated, faithful multiplication $R$ -module then for any two ideals $A$ and $B$ of $R$ , $AM\subseteq BM$ iff $A\subseteq B$ .

Proof of Theorem 1.4. $(\Rightarrow )$ By Theorem 1.2, the $R$ -module $R$ is a faithful, Noetherian, distributive $R$ -module.

$(\Leftarrow )$ Let $M$ be faithful, Noetherian, distributive $R$ -module.

(i) The ring $R$ is a Noetherian ring: The $R$ -module $M$ is Noetherian, hence finitely generated, $M=\sum _{i=1}^n Rm_i$ for some elements $m_1, \ldots, m_n\in M$ . The $R$ -module $M$ is a faithful module. Hence, the map $R\rightarrow \oplus _{i=1}^n Rm_i$ , $r\mapsto (rm_1, \ldots, rm_n)$ is an $R$ -monomorphism. The direct sum is a Noetherian $R$ -module (as a finite direct sum of Noetherian modules), and the statement (i) follows.

(ii) The ring $R$ has only finitely many minimal primes: The statement (ii) follows from the statement (i).

(iii) For all maximal ideals $\mathfrak{m}$ of the ring $R$ , the $R_{\mathfrak{m}}$ -module $M_{\mathfrak{m}}$ is faithful, Noetherian and distributive: The $R$ -module $M$ is finitely generated. Hence, $\textrm{ann}_{R_{\mathfrak{m}}}(M_{\mathfrak{m}}) = \textrm{ann}_R(M)_{\mathfrak{m}}=0$ since $\textrm{ann}_R(M)=0$ . Clearly, the $R_{\mathfrak{m}}$ -module $M_{\mathfrak{m}}$ is Noetherian and distributive (since the $R$ -module $M$ is so and localizations respect finite intersections).

(iv) The $R_{\mathfrak{m}}$ -module $M_{\mathfrak{m}}$ is a multiplication $R_{\mathfrak{m}}$ -module:

The statement (iv) follows from the statement (iii) and Theorem 2.7.(1).

(v) The R-module M is a multiplication module: The $R$ -module $M$ is finitely generated. By the statement (iv) and Theorem 2.7.(2), the $R$ -module $M$ is a multiplication $R$ -module.

Let ( $\mathcal{I} (R), \subseteq )$ be the lattice of ideals of the ring $R$ and $({\textrm{Sub}}_R(M), \subseteq )$ be the lattice of $R$ -submodules of the $R$ -module $M$ .

(vi) The map $\mathcal{I} (R)\rightarrow{\textrm{Sub}}_R(M)$ , $ I\mapsto IM$ is an isomorphism of latices: The $R$ -module $M$ is a finitely generated, faithful multiplication module (the statement (v)), and the statement (vi) follows from Theorem 2.7.(3).

(vii) The ring $R$ is a multiplication ring: The statement (vii) follows from the statements (v) and (vi).

Now, the theorem follows from the statements (ii) and (vii).

Proof of Theorem 1.5. $(\Rightarrow )$ See the statement (vi) in the proof of Theorem 1.4.

$(\Leftarrow )$ This implication follows at once from Theorem 1.3.