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«Duprime and Dusemiprime Modules John E. van den Berg University of Natal Pietermaritzburg, South Africa Robert Wisbauer University of D¨sseldorf, ...»

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Duprime and Dusemiprime Modules

John E. van den Berg

University of Natal Pietermaritzburg, South Africa

Robert Wisbauer

University of D¨sseldorf, Germany

u

Abstract

A lattice ordered monoid is a structure L; ⊕, 0L ; ≤ where L; ⊕, 0L is a

monoid, L; ≤ is a lattice and the binary operation ⊕ distributes over finite

meets. If M ∈ R-Mod then the set I M of all hereditary pretorsion classes of

L

σ[M ] is a lattice ordered monoid with binary operation given by

α :M β := {N ∈ σ[M ] | there exists A ≤ N such that A ∈ α and N/A ∈ β}, whenever α, β ∈ I M (the subscript in :M is omitted if σ[M ] = R-Mod). σ[M ] L is said to be duprime (resp. dusemiprime) if M ∈ α :M β implies M ∈ α or M ∈ β (resp. M ∈ α :M α implies M ∈ α), for any α, β ∈ I M. The main L results characterize these notions in terms of properties of the subgenerator M. It is shown, for example, that M is duprime (resp. dusemiprime) if M is strongly prime (resp. strongly semiprime). The converse is not true in general, but holds if M is polyform or projective in σ[M ]. The notions duprime and dusemiprime are also investigated in conjunction with finiteness conditions on I M, such as coatomicity and compactness.

L Introduction A classical example of lattice ordered monoid is given by the set of all ideals Id R of an arbitrary ring R with identity. Here, the lattice structure is induced by the relation of reverse set inclusion with ideal multiplication the binary operation. Several ring theoretic notions are characterizable as sentences in the language of the lattice ordered monoid Id R. Primeness and semiprimeness are two examples. An ideal P of a ring R is prime if and only if for any I, J ∈ Id R, IJ ⊆ P implies I ⊆ P or J ⊆ P, and semiprime if and only if for any I ∈ Id R, I 2 ⊆ P implies I ⊆ P.

Id R is embeddable in a larger lattice ordered monoid comprising the set of all hereditary pretorsion classes of R-Mod (denoted I R ) via the mapping L η : Id R → I R, I → η(I) := {M ∈ R-Mod | IM = 0}.

L The embedding η allows us to express the notions prime and semiprime, for example, in terms of hereditary pretorsion classes thus: P ∈ Id R is prime if and only if for all I, J ∈ Id R, η(I) : η(J) ⊇ η(P ) implies η(I) ⊇ η(P ) or η(J) ⊇ η(P ) and P is semiprime if and only if for all I ∈ Id R, η(I) : η(I) ⊇ η(P ) implies η(I) ⊇ η(P ). This observation motivates the introduction of a notion of ‘primeness’ and ‘semiprimeness’ in I R. We call γ ∈ I R dual prime, henceforth to be abbreviated duprime, if for all L L α, β ∈ I R, α : β ⊇ γ implies α ⊇ γ or β ⊇ γ, and γ is said to be dual semiprime, L henceforth dusemiprime, if for all α ∈ I R, α : α ⊇ γ implies α ⊇ γ. (The prefix L ‘dual’ is explained by the fact that the above sentence corresponds with the usual notion of primeness (resp. semiprimeness) if interpreted in the order dual of Id R.) Insofar as I R may be viewed as a structure which properly contains Id R (via L the embedding η), it is not difficult to see that P will be a prime ideal of R if η(P ) is duprime in I R. The latter condition is thus at least as strong as the former. In L particular, taking P to be the zero ideal, R will be a prime ring if the hereditary pretorsion class consisting of all left R-modules, namely R-Mod, is duprime in I R. L It is shown in [11, Theorem 26 and Remark 27] that the rings R for which R-Mod is duprime are precisely the left strongly prime rings of Handelman and Lawrence [7].

It is shown similarly [11, Theorem 32 and Remark 33] that R-Mod is dusemiprime if and only if R is left strongly semiprime in the sense of Handelman [6].

Viewing R-Mod as the hereditary pretorsion class subgenerated by the module R R, these results can be seen as an attempt to characterize duprimeness and dusemiprimeness of σ[R R] in terms of properties of the subgenerator R R. This paper addresses the following natural generalization: if M is an arbitary module, characterize duprimeness and dusemiprimeness of the hereditary pretorsion class σ[M ] in terms of properties of the subgenerator M.

Results do not generalize easily from R R to a general M, for the module R R is finitely generated and projective. These rather special properties impart a type of finiteness to R-Mod which is absent in the case of a general σ[M ]. Every strongly prime module, in the sense of [1], subgenerates a duprime hereditary pretorsion class.

But the converse turns out to be false, in general.

Results in this paper have a mixed flavour; they make use of standard module theoretic techniques, but are also reliant on the body of theory on lattice ordered monoids developed in [11].

1 Preliminaries The symbol ⊆ denotes containment and ⊂ proper containment for sets. Throughout the paper R will denote an associative ring with identity, R-Mod the category of unital left R-modules, and M any object in R-Mod. If N is a submodule (resp.

essential submodule) of M we write N ≤ M (resp. N ¢ M ). We denote the left annihilator of a subset X of M by (0 : X). We call M cofaithful if (0 : X) = 0 for some finite subset X of M.

–  –  –

where Tr(σ[M ], N ) denotes the trace of the class σ[M ] in N. Tr(σ[M ], N ) corresponds with the unique largest submodule of N contained in σ[M ]. It follows from properties of injectives that Tr(σ[M ], N ) = Tr(M, N ) whenever N is injective in σ[M ].





The collection of all hereditary pretorsion classes of R-Mod is a set [9, Proposition VI.4.2, p. 145] whose elements we shall denote by α, β,..., or by σ[M ] if we wish to refer to a specific subgenerator. We shall, for notational convenience, identify a hereditary pretorsion class α with its associated left exact preradical and write α(N ) in place of Tr(α, N ) whenever N ∈ R-Mod. We call K ≤ M a pretorsion submodule of M if K = α(M ), for some hereditary pretorsion class α. Every pretorsion submodule of M is fully invariant in M. If M is injective in σ[M ], then the converse is also true, for if U is a fully invariant submodule of M and α = σ[U ], then α(M ) = U.

1.2 The Grothendieck category σ[M ]. Coproducts, quotient objects and subobjects in σ[M ] are the same as in R-Mod because of the defining closure properties of a hereditary pretorsion class [18, 15.1((1),(2)), p. 118]. It follows that the hereditary pretorsion classes of σ[M ] are precisely the hereditary pretorsion classes of R-Mod which are contained in σ[M ]. For the most part, these shall be our objects of study.

Put α = σ[M ]. If {Ni | i ∈ Γ} is a family of modules in α then α Ni := α( i∈Γ Ni ) i∈Γ is the product of {Ni | i ∈ Γ} in α [18, 15.1(6), p. 118], and E α (N ) := α(E(N )) is the injective hull of N in α [18, 17.9(2), p. 141]. If A is a nonempty class of modules

in R-Mod we introduce two abbreviations:

Pα (A) = {N ∈ α | N = α( i∈Γ Ai ), for some family {Ai | i ∈ Γ} in A}, Eα (A) = {N ∈ α | N = α(E(A)) for some A ∈ A}.

We claim that α ∩ SPE(A) = SPα Eα (A).

Since SPE(A) is a torsion-free class in R-Mod containing A, it follows that SPα Eα (A) ⊆ SPE(A). The containment in one direction follows. The reverse containment follows since α ∩ SPE(A) ⊆ SPα E(A) = SPα Eα (A). Observe that if A ⊆ α then α ∩ SPE(A) = SPα Eα (A) is the smallest torsion-free class of α containing A.

–  –  –

I M ; :M, {0}; ⊆ is thus an integral lattice ordered monoid for all M.

L We warn the reader that, inasmuch as the operations :M and ‘:’ differ, an idempotent element of I M, i.e., hereditary torsion class of σ[M ], need not be idempotent L in I R.

L

–  –  –

1.7 Special subgenerators. Since σ[M ] is a Grothendieck category we can always find an injective subgenerator for σ[M ], for example the M -injective hull M of M.

We can even find an injective cogenerator for σ[M ] which is also a subgenerator for σ[M ], for example M ⊕ Q, where Q is any injective cogenerator for σ[M ]. Notice that not every cogenerator is a subgenerator. For example, Q/Z is an injective IZ cogenerator but not a subgenerator for Z Z-Mod since σ[Q/Z is just the class of I Z] torsion ZZ-modules.

Let Q ∈ R-Mod be injective in σ[M ]. We call Q a big cogenerator for σ[M ] if SC({Q}) contains all finitely generated modules in σ[M ]. Big cogenerators are important because they are both subgenerators and cogenerators. The former property is a consequence of the fact that the hereditary pretorsion class HSC({Q}) ⊇ SC({Q}) contains all finitely generated modules in σ[M ], whence HSC({Q}) = σ[M ]. To see the latter property, observe that if Q is a big cogenerator for α = σ[M ], then the torsion-free class SPE({Q}) of R-Mod contains every finitely generated module in α, whence SPE({Q}) ⊇ α and so α = α ∩ SPE({Q}) = SPα Eα ({Q}).

Since Q is injective in α, α = SPα ({Q}) = SPα Eα ({Q}). We conclude that Q is a cogenerator for σ[M ].

For example, if M is locally noetherian, then the direct sum of a representative set of indecomposable (uniform) injective modules in σ[M ] is an (injective) big cogenerator for σ[M ]. If M is locally of finite length (i.e., locally artinian and noetherian) then every injective cogenerator for σ[M ] is a big cogenerator for σ[M ].

–  –  –

where Ke f = Ke fK ∩ Ke fL = 0.

Since M is projective in σ[M ] (or polyform) and K is M -singular, Ke fK ¢ M.

This implies Ke fL = 0 and M ∈ σ[L].

We point out that the polyform case (δM = λM ) also follows from Lemma 2.1. P It is an elementary fact that for any ideal I of R, R/I-Mod = R-Mod if and only if I = 0. If U is a fully invariant submodule of M then the statement σ[M/U ] = σ[M ] implies U = 0, does not hold in general. It does, however, hold if M is projective in σ[M ], as shown in [17, Lemma 2.8, p. 3623]. Lemma 2.4 below identifies another condition sufficient for the implication to hold. We first require a preliminary result.

Recall that the smallest hereditary torsion class of σ[M ] containing δM is called the M -Goldie torsion class. It is shown in [19, 10.5, p. 74] that the M -Goldie torsion class coincides with δM :M δM.

2.3 Lemma. Let U = α(M ) where α is a hereditary torsion class of σ[M ]. If σ[M/U ] = σ[M ], then U belongs to the M -Goldie torsion class.

–  –  –

2.4 Lemma. Suppose M is polyform and U = α(M ) for some hereditary torsion class α of σ[M ]. Then σ[M ] = σ[M/U ] if and only if U = 0.

Proof. The implication in one direction is obvious. Suppose σ[M ] = σ[M/U ] and let γ denote the M -Goldie torsion class. By Lemma 2.3, U ∈ γ. But M is polyform so δM (M ) = 0. Since δM and γ have the same associated torsion-free class, we must have γ(U ) ⊆ γ(M ) = 0. We conclude that U = 0, as required. P 3 Duprime modules Interpreting [11, Theorem 14] in the case where the lattice ordered monoid is chosen

to be I M, we obtain:

L

3.1 Theorem. The following assertions are equivalent for a left R-module M :

–  –  –

The results which follow reveal a rich variety of characterizations of duprime modules in the case where a finiteness condition is imposed on the lattice I M.

L Recall that M is said to be strongly prime if α(M ) = 0 or α(M ) = M for all α ∈ I R. The study of strongly prime modules was initiated in Beachy-Blair [1].

L It is clear from the definition that M will be strongly prime if and only if every proper element of I M is contained in λM. Further characterizations of strongly L prime modules may be found in [19, 13.3, p. 96].

It is an immediate consequence of Theorem 3.1 that every strongly prime module is duprime. In Example 3.4 we exhibit a module which is duprime but not strongly prime. Thus duprimeness is a strictly weaker notion. The reader will observe that the duprimeness of M depends only on properties of the lorrim I M, and in fact, if M L is duprime then every subgenerator of σ[M ] inherits the same property. In contrast, strong primeness is an intrinsic property of the module M. If M is strongly prime it is not necessarily the case that every subgenerator for σ[M ] is strongly prime.

However, as Theorem 3.3 shows, if M is duprime then every projective or polyform subgenerator for σ[M ] is strongly prime.

3.2 Theorem. The following assertions are equivalent for a nonzero left R-module

M:

–  –  –

In general, the conditions: (1) M is projective in σ[M ], and (2) M is polyform, are independent. If M is duprime then condition (1) is stronger than (2). To see this, suppose M is duprime and projective in σ[M ]. Note that σ[M/δM (M )] = {0} or σ[M ] by Theorem 3.1(e). The former implies M/δM (M ) = 0, whence M ∈ δM.

But this contradicts the fact that δM is small in I M (Lemma 2.2). Consequently, we L must have σ[M/δM (M )] = σ[M ]. This implies δM (M ) = 0, i.e., M is polyform, as noted in the discussion preceding Lemma 2.3. In Section 4 we shall improve on this result by showing that (1) implies (2) under conditions weaker than M duprime.

3.3 Theorem. Assume M is projective in σ[M ] or M is polyform. Then the following assertions are equivalent:

(a) M is duprime;

(b) M is strongly prime.

Proof. (b)⇒(a) holds with no assumption on M.

(a)⇒(b) Since M is by hypothesis duprime, M will be polyform if M is projective in σ[M ]. It therefore suffices to establish (b) in the case where M is polyform.

Suppose U = α(M ) is a proper pretorsion submodule of M. To establish the strong primeness of M we must show that U = 0. Since U is a proper submodule of M we cannot have M ∈ σ[U ]. It follows from Theorem 3.1(h), that M ∈ σ[M/U ].

Assume U is essential in M. Then M/U is M -singular, but M is non-M -singular, so we cannot have M ∈ σ[M/U ], a contradiction. We conclude that U is not essential in M. Let α denote the unique smallest hereditary torsion class containing α.



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