Explicit Description of a Class of Indecomposable Injective Modules^{*}

M. R. Pournaki, M. Tousi

Abstract Let R be a commutative Noetherian ring and 𝔭 be a prime ideal of R such that the ideal 𝔭R_{𝔭} is principal and ht(𝔭) ≠ 0. In this note, we describe the explicit structure of the injective envelope of the R-module R/𝔭.

Keywords: Noetherian ring, Injective module, Indecomposable injective module, Injective envelope, Weakly locally principal prime ideal. 2000 Mathematics Subject Classification: Primary 13C11, 13B30; Secondary 13E05.

1 Introduction

According to classic results of E. Matlis [5, Theorems 2.5 and 2.7] every injective module over a Noetherian ring R can be expressed uniquely as the direct sum of indecomposable injective modules; the indecomposables have the form E_{R}(R/J) where J is an irreducible left ideal of R [5, Theorem 2.4]; and if in addition R is commutative the indecomposables are exactly the envelopes E_{R}(R/𝔭), 𝔭 being a prime ideal of R [5, Proposition 3.1]. Thus if we wish to understand the structure of the injective modules in detail, it suffices to know the structure of the indecomposables. Finding a precise description of a class of indecomposable injective modules was the main object of [7], [2], [4], [10], [9], and [1], although even over commutative Noetherian ring their structure can still be quite complicated.

In this note we give the explicit structure of the injective envelope of the Rmodule R/𝔭, where 𝔭 is a prime ideal of R with ht(𝔭) ≠ 0 and is weakly locally principal, i.e., a prime ideal of R such that there exists an element p of R for which 𝔭R_{𝔭} = pR_{𝔭}. Note that the prime ideals 𝔭 with ht(𝔭) = 1 in regular rings, Krull rings and Noetherian normal rings are weakly locally principal. In particular, each prime ideal of Dedekind domain is weakly locally principal too.

*This research was in part supported by a grant from IPM.

2 Main Results

Throughout this section, let R denote a commutative ring with identity, M be a unitary left R-module and E_{R}(−) denote the injective envelope of R-module −. Also, if 𝔭 denotes the weakly locally principal prime ideal of R, then p denotes the element of R for which 𝔭R_{𝔭} = pR_{𝔭}. For such 𝔭 and p, define S = {p^{i}s : s ∈ R \ 𝔭, i ≥ 0}. Clearly S is a multiplicative closed subset of R and we have R \ 𝔭 ⊆ S. In this case, for such S, the function Θ : R_{𝔭} ⟶ S^{−1}R defined by Θ(r/s) = r/s is an R-homomorphism.

In the following theorem, the explicit structure of a class of indecomposable injective modules will be given.

Main Theorem Let R be a Noetherian ring and 𝔭 a weakly locally principal prime ideal of R. If ht(𝔭) ≠ 0, then E_{R}(R/𝔭) ≅ S^{−1}R/Θ(R_{𝔭}) as R-modules.

Let 𝔞 be an ideal of R. For each R-module M, set Γ_{𝔞}(M) = _{⋃n∈ℕ}(0 :_{M} 𝔞^{n}), the set of elements of M which are annihilated by some power of 𝔞.

For the proof of the Main Theorem we need to prove the following lemmas.

Lemma 2.1 Let M and E be R-modules and E be injective. If a is an element of R such that aM = M, Γ_{aR}(M) = M, Γ_{aR}(E) = E and (0 :_{M} a) ≅ (0 :_{E} a), then M ≅ E.

Proof. By hypothesis, there is an R-isomorphism φ : (0 :_{M} a) ⟶ (0 :_{E} a) and therefore we obtain the induced R-monomorphism φ̂ : (0 :_{M} a) ⟶ E. Now, injectivity of E implies that there is an R-homomorphism ψ : M ⟶ E such that ψ|_{(0:_{M} a)} = φ̂. We claim that ψ is an R-isomorphism.

If K is an R-module such that Γ_{aR}(K) = K, then for x ∈ K \ {0} we define exp(x) = min{n ∈ ℕ : a^{n}x = 0} and we set exp(0) = 0.

ψ is injective: We show that x ∈ Ker ψ implies x = 0. We use induction on exp(x). If x ∈ Ker ψ and exp(x) = 1, then ax = 0, so x ∈ (0 :_{M} a). Therefore 0 = ψ(x) = φ̂(x) and so x = 0. Now suppose, inductively, x ∈ Ker ψ, exp(x) = n > 1 and suppose for each y ∈ Ker ψ with exp(y) = n − 1, we have shown that y = 0. The condition exp(x) = n implies that exp(ax) = n − 1. But ax ∈ Ker ψ, so, by the inductive hypothesis, ax = 0. Since n > 1, we have x = 0. This completes the inductive step.

ψ is surjective: Again we use induction. Suppose y ∈ E and exp(y) = 1. Then ay = 0 and we have y ∈ (0 :_{E} a). Now surjectivity of φ implies that there is x ∈ (0 :_{M} a) ⊆ M, such that y = φ(x) = φ̂(x) = ψ(x), so y ∈ Im ψ. Now suppose, inductively, y ∈ E, exp(y) = n > 1 and suppose for each z ∈ E with exp(z) = n − 1, we have shown that z ∈ Im ψ. Since y ∈ E and exp(y) = n implies that exp(ay) = n − 1, by the inductive hypothesis there is x′ ∈ M such that ψ(x′) = ay. Since aM = M, x′ = ax′_{a}, where x′_{a} ∈ M. Now we have ψ(x′_{a}) = y or exp (ψ(x′_{a}) − y) = 1. In either case we have y ∈ Im ψ. This completes the inductive step.

Therefore we establish the claim and so M ≅ E. □

Lemma 2.2 Let R be a Noetherian ring and 𝔭 be a weakly locally principal prime ideal of R for which ht(𝔭) ≠ 0. Then R_{𝔭}/𝔭R_{𝔭} ≅ (0 :_{S^{−1}R/Θ(R_{𝔭})} p) as R-modules.

Proof. Define ϕ : R_{𝔭}/𝔭R_{𝔭} ⟶ (0 :_{S^{−1}R/Θ(R_{𝔭})} p) by ϕ(r/s+𝔭R_{𝔭}) = r/sp+Θ(R_{𝔭}). Clearly ϕ is an R-homomorphism. Firstly, we prove that ϕ is surjective. For showing this, suppose α/p^{i}t + Θ(R_{𝔭}) ∈ (0 :_{S^{−1}R/Θ(R_{𝔭})} p). Therefore, there exist t′ ∈ R \ 𝔭 and β ∈ R for which pα/p^{i}t = β/t′ or α/p^{i}t = β/pt′ in S^{−1}R. Now, ϕ(β/t′ + 𝔭R_{𝔭}) = β/pt′ + Θ(R_{𝔭}) = α/p^{i}t + Θ(R_{𝔭}) implies that ϕ is surjective. Secondly, we claim that ϕ is injective. Suppose the contrary, i.e., there is a non-zero element in Ker ϕ, say r/s + 𝔭R_{𝔭}. So there exists r′/s′ ∈ Θ(R_{𝔭}) such that r/sp = r′/s′ in S^{−1}R and r/s ∉ 𝔭R_{𝔭}. Therefore there exists l ≥ 0 and t ∈ R \ 𝔭 for which p^{l}trs′ = p^{l+1}tr′s. Consequently (𝔭R_{𝔭})^{l+1} = (𝔭R_{𝔭})^{l}. Nakayama Lemma now implies that ht(𝔭R_{𝔭}) = 0, a contradiction. So the claim is proved and ϕ is an R-isomorphism and the lemma holds. □

Proof of the Main Theorem.

Using [8, Lemma 4.24], we get (0 :_{E_{R}(R/𝔭)} p) ≅ (0 :_{E_{R𝔭}(R_{𝔭}/𝔭R_{𝔭})} 𝔭R_{𝔭}) ≅ R_{𝔭}/𝔭R_{𝔭}. On the other hand, Lemma 2.2 implies that R_{𝔭}/𝔭R_{𝔭} ≅ (0 :_{S^{−1}R/Θ(R_{𝔭})} p). Therefore we have

(0 :_{E_{R}(R/𝔭)} p) ≅ (0 :_{S^{−1}R/Θ(R_{𝔭})} p).

It is easy to see that p(S^{−1}R/Θ(R_{𝔭})) = S^{−1}R/Θ(R_{𝔭}). Now, suppose r/p^{i}s+Θ(R_{𝔭}) ∈ S^{−1}R/Θ(R_{𝔭}). Therefore, p^{i}(r/p^{i}s + Θ(R_{𝔭})) = Θ(R_{𝔭}) and so r/p^{i}s + Θ(R_{𝔭}) ∈ Γ_{pR}(S^{−1}R/Θ(R_{𝔭})). This shows that Γ_{pR}(S^{−1}R/Θ(R_{𝔭})) = S^{−1}R/Θ(R_{𝔭}). Since Γ_{pR}(E_{R}(R/𝔭)) = E_{R}(R/𝔭) (see [6, Theorem 18.4 (v), (vi)]), Lemma 2.1 implies that

E_{R}(R/𝔭) ≅ S^{−1}R/Θ(R_{𝔭}). □

Let p be an element of R and λ : R ⟶ R_{p} be natural R-homomorphism. Then we denote the R-module R_{p}/λ(R)={a/p^{n} + λ(R) : a ∈ R, n ≥ 0} by R_{p^{∞}}.

Proposition 2.3 Let p be an element of R such that 𝔭 = pR is a maximal ideal of R. Then 𝔭 is the weakly locally principal prime ideal of R and if we consider S = {p^{i}s : s ∈ R \ 𝔭, i ≥ 0} and Θ : R_{𝔭} ⟶ S^{−1}R as we mentioned earlier we obtain S^{−1}R/Θ(R_{𝔭}) ≅ R_{p^{∞}} as R-modules.

Proof. For each r ∈ R \ pR and each l ≥ 0, there exists α, β ∈ R such that αr + βp^{l} = 1. By using this fact, it is easy to see that the natural R-homomorphism μ : R_{p^{∞}} ⟶ S^{−1}R/Θ(R_{𝔭}) given by μ(r/p^{n} + λ(R)) = r/p^{n} + Θ(R_{𝔭}) is an Risomorphism. □

Now the Main Theorem and Proposition 2.3 imply the following corollaries.

Corollary 2.4 Let R be a Noetherian ring and 𝔭 = pR be a maximal ideal of R. If ht(𝔭) ≠ 0, then E_{R}(R/𝔭) ≅ R_{p^{∞}} as R-modules.

Corollary 2.5 Let R be a Noetherian integral domain. If pR is a non-zero maximal ideal of R, then E_{R}(R/pR) ≅ R_{p^{∞}} as R-modules. In particular, if p is a prime integer, then E_{ℤ}(ℤ/pℤ) ≅ ℤ_{p^{∞}} as ℤ-modules.

We now apply the result of Corollary 2.5 to find a decomposition for injective modules over one-dimensional unique factorization domains. In the following, μ(−, M) denotes the 0-th Bass number of M with respect to prime ideal −. For an R-module N, ⨁ μ(−, M)N denotes the direct sum of μ(−, M) copies of N and consider Π = {p ∈ R \ {0} : pR ∈ Ass_{R}(M)}, where Ass_{R}(M) = {𝔭 ∈ Spec(R) : there exists x ∈ M such that 𝔭 = (0 :_{R} x)}.

Corollary 2.6 Let R be a one-dimensional unique factorization domain, F its field of fractions and let M be an injective R-module. Then

M ≅ ( ⨁ μ(0, M)F ) ⊕ ( ⨁_{p∈Π} μ(pR, M)R_{p^{∞}})

as R-modules.

We need the following lemma to prove this corollary.

Lemma 2.7 The ring R is a principal ideal domain if and only if R is a onedimensional unique factorization domain.

Proof. Clearly any principal ideal domain is a unique factorization domain and one-dimensional, so we prove the converse which is more interesting. Suppose that R is a one-dimensional unique factorization domain. We note that the proof of Theorem 20.1 in [6] shows that if R is a unique factorization domain and 𝔭 is a prime ideal of R such that ht(𝔭) = 1, then 𝔭 is a principal ideal. Since R is onedimensional, any prime ideal of R is principal. So R is Noetherian. Now if R is not a principal ideal domain, there is a non-principal ideal 𝔞. Since R is Noetherian, there is an ideal, 𝔪, that is maximal with respect to being non-principal. A standard result of M. Isaacs (see [3, page 8]) states that 𝔪 is a prime ideal. This contradicts the fact that all prime ideals of R are principal and completes the proof. □

Proof of the Corollary 2.6.

Let 0 ≠ 𝔭 ∈ Ass_{R}(M). By Lemma 2.7, 𝔭 = pR for some p ∈ Π. We know that M has a decomposition in the form

M ≅ ⨁_{𝔭∈Spec(R)}μ(𝔭, M)E_{R}(R/𝔭)

(see [5, Theorem 2.5 and Proposition 3.1]). But it is easy to see that μ(𝔭, M) ≠ 0 if and only if 𝔭 ∈ Ass_{R}(M). Therefore since E_{R}(R) ≅ F we have

M ≅ _{⨁𝔭∈Ass_{R}(M)}μ(𝔭, M)E_{R}(R/𝔭)
≅ ( ⨁ μ(0, M)E_{R}(R)) ⊕ (_{⨁0≠𝔭∈Ass_{R}(M)}μ(𝔭, M)E_{R}(R/𝔭))
≅ ( ⨁ μ(0, M)F ) ⊕_{( ⨁p∈Π} μ(pR, M)R_{p^{∞}}). □

Acknowledgment: This work was done while the first author was a Postdoctoral Research Associate at the School of Mathematics, Institute for Studies in Theoretical Physics and Mathematics (IPM). He would like to thank the IPM for the financial support. The authors also thank Irena Swanson and Jim Coykendall for their useful comments which led to the improvement of the first draft.

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The Authors’ Addresses

M. R. Pournaki, School of Mathematics, Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-5746, Tehran, Iran.

E-mail address: pournaki@ipm.ir

M. Tousi, School of Mathematics, Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-5746, Tehran, Iran, and Department of Mathematics, Shahid Beheshti University, Evin, Tehran 19839, Iran. E-mail address: mtousi@ipm.ir