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 [Reference Krull5]. In [Reference Mott6], 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.
([Reference Alsuraiheed and Bavula1
, 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 [Reference Alsuraiheed and Bavula1]. It is given via the intersection and strong orthogonality conditions.
Theorem 2.4.
([Reference Alsuraiheed and Bavula2]) 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. The endomorphism ring
${ \textrm{End }}_R(M)$
is also a multiplication ring.
-
2.
${ \textrm{End }}_R(M)\simeq R/\textrm{ann}_R(M)$
. -
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. ([Reference Barnard3 , 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. ([Reference Barnard3 , 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. ([Reference Barnard4 , 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).
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