(To appear in: Rendiconti del Seminario Mathematico della Universita di Padova, vol 97, 1997)

Minimal non-totally minimal topological rings

M.G. Megrelishvili (Levy)* Department of Mathematics and Computer Science Bar-Ilan University, 52 900 Ramat-Gan, ISRAEL

Abstract. We establish the existence of minimal non-totally minimal topological rings with a unit answering a question of Dikranjan. The Pontryagin duality and a generalization of Ursul’s “semidirect product type” construction play major roles in the construction.

Introduction. A Hausdorff topological ring R is called minimal if its topology is minimal in the sense of Zorn among all Hausdorff ring topologies on R. If R/J is minimal for every closed ideal J, then R is called totally minimal [2].

The induced topology of a nontrivial valuation on a field is (totally) minimal (see [10, 6]). Some generalizations and related results in the context of fields or divisible rings may be found in [11, 13, 14]. For more general cases we refer to [1, 2, 3, 9]. Recall [2, 3] for instance that the class of all minimal rings with a unit is closed under forming topological products, direct sums and matrix rings. If P is a non-zero prime ideal of finite index in a Dedekind ring, then the P -adic topology is minimal.

The question about existence of minimal non-totally minimal rings with a unit is discussed by Dikranjan in [2, 3].

Conventions and preliminaries. As usual ℕ, ℤ, ℝ denote the set of all natural, integer and real numbers, respectively. The unit circle group ℝ/ℤ will be denoted by 𝕋 and the n-element cyclic ring by ℤ_{n}.

All rings are assumed to be associative. A ring R is unital if it has a unit. The zero-element will be denoted by 0. By char(R) we indicate the minimal natural number (if it exists) n such that nx = 0 for every x ∈ R. Otherwise we write char(R) = 0. Clearly, char(R) = n > 0 iff R is a (left) ℤ_{n}-algebra in a natural way (k, x) ↦ x + x + ⋯ + x (k terms) for each (k, x) ∈ ℤ_{n} × R.

Typeset by AMS-TEX

*Partially supported by the Israel Ministry of Sciences, Grant No. 3505.

1991 Mathematics Subject Classification. 16W80, 54H13, 13J99, 22B99.

Key words and phrases. Minimal ring, locally compact ring, Pontryagin duality.

For a locally compact Abelian group G, denote by G^{∗} the dual group H(G, 𝕋) of all continuous characters endowed with the compact open topology. If R is a locally compact ring, then R^{∗} is a topological (R, R)-bimodule [12].

If P is a subgroup of a topological group (G, τ), then τ |_{P} will denote the relative topology on P, and τ/P will be the quotient topology on the left coset space G/P . The following useful result is well known.

Merzon’s Lemma. [8] (See also [4, Lemma 7.2.3] for a proof).

Let P be a subgroup of a group G, and let τ′ and τ be (not necessarily Hausdorff) group topologies on G with the properties : τ′ ⊆ τ, τ′ |_{P} = τ |_{P} and τ′/P = τ/P. Then τ′ = τ.

Main results. Recall a construction from [12]. Let R be a topological ring and X a topological (R, R)-bimodule. On the product R × X of topological groups R and X, consider the multiplication

(r_{1}, x_{1}) (r_{2}, x_{2}) = (r_{1}r_{2}, r_{1}x_{2} + x_{1}r_{2}), r_{1}, r_{2} ∈ R , x_{1}, x_{2} ∈ X.

Then R × X becomes a topological ring which is denoted by R ⋌ X. For details and a particular case of R ⋌ R^{∗} see Ursul [12].

Now we generalize this construction in two directions. The first change is minor. Let K be a commutative unital Hausdorff topological ring, (R, τ) a topological K-algebra, and (S, ν) be a topological K-module. Instead of R^{∗} = H_{ℤ}(R, 𝕋), consider the K-module H_{K}(R, S) of all continuous K-homomorphisms R → S. As in the case of R^{∗}, the left and right multiplications in R induce the (R, R)- bimodule structure in H_{K}(R, S). The second modification is more essential. We add to R ⋌ H_{K}(R, S) a supplementary coordinate. Denote by M_{K}(R, S) the product R × H_{K}(R, S) × S of K-modules. The multiplication we define by the rule:

(r_{1}, f_{1}, s_{1}) (r_{2}, f_{2}, s_{2}) = (r_{1}r_{2}, r_{1}f_{2} + f_{1}r_{2}, f_{2}(r_{1}) + f_{1}(r_{2}))

where r_{1}, r_{2} ∈ R, f_{1}, f_{2} ∈ H_{K}(R, S) and s_{1}, s_{2} ∈ S. Simple computations show that M_{K}(R, S) becomes a K-algebra. Let H_{K}(R, S) carry a K-module topology σ such that its (R, R)-bimodule structure is topological too. Moreover, suppose that the evaluation mapping

ω: H_{K}(R, S) × R ⟶ S, ω(f, r) = f(r)

is continuous with respect to the triple (σ, τ, ν) of Hausdorff topologies. Then (M_{K}(R, S), γ) is a Hausdorff topological K-algebra with respect to the product topology γ. In particular, if R is a locally compact ring, S = 𝕋 and K = ℤ, then one gets a topological ring M_{ℤ}(R, 𝕋) = R×R^{∗} ×𝕋 which will be denoted by M(R).

Furthermore, we identify R, H_{K}(R, S) and H_{K}(R, S)×S with the corresponding subsets of M_{K}(R, S). We will keep below our assumptions about (M_{K}(R, S), γ).

Proposition 1. Let γ′ be a new ring topology on M_{K}(R, S) such that the canonical group retraction q: H_{K}(R, S)×S ⟶ S is continuous for the topologies γ′ |_{H_{K}(R,S)×S} and ν.Then the evaluation mapping ω is continuous with respect to the triple of topologies γ′ |_{H_{K}(R,S)} , γ′/H_{K}(R, S) × S and ν.

Proof. Fix φ_{0} ∈ H_{K}(R, S), r_{0} ∈ R and a ν-neighborhood O at φ_{0}(r_{0}) in S. By the continuity of q, we may choose a γ′-neighborhood U of the element z_{0} = (0, φ_{0}r_{0}, φ_{0}(r_{0})) ∈ M_{K}(R, S), such that q(U ∩ (H_{K}(R, S) × S)) ⊆ O.

By our assumption, the ring multiplication is γ′-continuous. Therefore, there exist γ′-neighborhoods V, W of the elements (0, φ_{0}, 0) and (r_{0}, 0, 0) respectively, such that V ·W is contained in the chosen γ′-neighborhood U of z_{0} = (0, φ_{0}, 0) (r_{0}, 0, 0).

For every φ ∈ V ∩ H_{K}(R, S) and every (r, f, s) ∈ W, we have

(0, φ, 0) (r, f, s) = (0, φr, φ(r)) ∈ U ∩ (H_{K}(R, S) × S).

Clearly, φ(r) = ω(φ, r) ∈ ω(V ∩H_{K}(R, S), pr(W)), where pr denotes the projection M_{K}(R, S) ⟶ R on the first coordinate.

Then,

φ(r) ∈ ω(V ∩ H_{K}(R, S), pr(W)) ⊆ q(U ∩ (H_{K}(R, S) × S)) ⊆ O.

Since pr(W) is a γ′/H_{K}(R, S) × S-neighborhood of the point r_{0} and V ∩H_{K}(R, S) is a γ′ |_{H_{K}(R,S)}-neighborhood of the point φ_{0}, then the continuity of ω at (φ_{0}, r_{0}) is proved. □

Let (F, σ). (E, τ), (S, ν) be Abelian Hausdorff groups. A continuous mapping ω: F ×E ⟶ S is called biadditive if the induced mappings ω_{x}: F ⟶ S, ω_{f}: E ⟶ S are homomorphisms for every x ∈ E and every f ∈ F. We say that a coarser pair (σ′, τ′) ≤ (σ, τ) of group topologies is ω-compatible if ω remains continuous with respect to the triple (σ′, τ′, ν). If ω is separated (i.e., if the annihilators of E and F are both zero), then the Hausdorff property of ν implies that every ω-compatible pair (σ′, τ′) is necessarily Hausdorff. Following [7], we say that ω is minimal if for every ω-compatible pair (σ′, τ′) ≤ (σ, τ), we have necessarily σ′ = σ, τ′ = τ.

Lemma 2. [7, Proposition 1.10]. For every Hausdorff locally compact Abelian group G, the evaluation mapping G^{∗} × G ⟶ 𝕋 is minimal.

Another example of a minimal biadditive mapping is the canonical duality E^{∗} × E ⟶ ℝ for a normed space E. Proposition 3. Let the evaluation mapping

ω: (H_{K}(R, S), σ) × (R, τ) ⟶ (S, ν)

be minimal, and let γ′ ⊆ γ, be a coarser Hausdorff ring topology on M_{K}(R, S) such that γ′ and γ coincide on H_{K}(R, S) × S. Then γ′ = γ.

Proof. Because γ′ and γ agree on H_{K}(R, S) × S, then, in particular, the mapping

q: H_{K}(R, S) × S ⟶ S

is continuous with respect to the pair (γ′ |_{H_{K}(R,S)×S} , ν). So, we can apply Proposition 1. Then γ′ |_{H_{K}(R,S)} , γ′/H_{K}(R, S) × S is a ω-compatible pair of group topologies. The minimality of ω implies γ′/H_{K}(R, S) × S = τ = γ/H_{K}(R, S) × S. Now Merzon’s Lemma finishes the proof. □

As a corollary we get Proposition 4. Let the evaluation mapping ω be minimal and let S and H_{K}(R, S) be compact. Then M_{K}(R, S) is a minimal ring.

Theorem 5. Let R be a discrete ring. Then the topological ring M(R) = R × R^{∗} × 𝕋 is minimal. Hence, every (commutative) discrete ring is a continuous ring retract of a minimal (commutative) locally compact ring.

Proof. By Pontryagin’s Theorem, R^{∗} is compact iff R is discrete. Now the minimality of M(R) follows from Lemma 2 and Proposition 4. The canonical retraction pr: M(R) ⟶ R is the desired one. □

The ring M(R) from Theorem 5 is not unital. In order to “improve” this, we use a well known unitalization procedure. Let R be a topological K-algebra. Consider a new K-algebra

R_{+} = {r + α1_{+} | r ∈ R, α ∈ K}

adjoining a unit 1_{+}. More precisely, R_{+} is a topological K-module sum R ⊕ K, and we identify (r, α) = r + α1_{+}. A multiplication on R_{+} is defined in the following manner:

(a + α1_{+}) (b + β1_{+}) = ab + αb + βa + αβ1_{+}

where α, β ∈ K and a, b ∈ R. The following lemma is trivial.

Lemma 6. If J is a (closed) ideal in R, then J is a (closed) ideal in R_{+} and R_{+}/J = (R/J)_{+}.

In the following result we use a method familiar from the theory of minimal topological groups (see, for example, [5]).

Theorem 7. Let R be a complete K-algebra such that (R, τ), (K, σ) are minimal topological rings. Then the K-unitalization R_{+} is a minimal topological ring.

Proof. Denote by γ the given product topology on R_{+} and suppose that γ′ ⊆ γ is a new Hausdorff ring topology. Since (R, τ) is a minimal ring, γ′ |_{R}= γ |_{R}= τ. By our assumption, (R, τ) is complete. Therefore, R is a closed ideal in (R_{+}, γ′). Consider the Hausdorff ring topology γ′/R on K. Since γ′/R ⊆ γ/R = σ and (K, σ) is a minimal ring, then γ′/R = γ/R. By Merzon’s Lemma we get γ′ = γ. □

Corollary 8. Let R be a minimal complete ring with char(R) = n > 0. Then the ℤ_{n}-unitalization R_{+} of R is a minimal ring.

Theorem 9. Let R be a discrete ring with char(R) = n > 0. Then the ℤ_{n}- unitalization R_{+} of R is a continuous ring retract of a minimal locally compact unital ring M_{+}.

Proof. Apply our construction for the situation S = K = ℤ_{n} and consider the ℤ_{n}-algebra M: = M_{ℤ_{n}}(R, ℤ_{n}) = R × H_{Z_{n}}(R, ℤ_{n}) × ℤ_{n}. Denote by M_{+} the ℤ_{n}- unitalization of M. Since char(R) = n > 0, then every character ξ: R ⟶ 𝕋 can actually be considered as a restricted homomorphism R ⟶ ℤ_{n} ⊂ 𝕋 identifying ℤ_{n} with the n-element cyclic subgroup of 𝕋. It is also clear that every homomorphism R ⟶ ℤ_{n} is even a morphism of ℤ_{n}-algebras. Therefore, H_{ℤ_{n}}(R, ℤ_{n}) and R^{∗} = H(R, 𝕋) coincide algebraically. Endow H_{ℤ_{n}}(R, ℤ_{n}) with the compact topology σ of R^{∗}. Eventually, the mapping ω: (R, τ) × (H_{ℤ_{n}}(R, ℤ_{n}), σ) ⟶ ℤ_{n} ⊂ 𝕋 is minimal, because of Lemma 2. By Proposition 4, the ring M is minimal. Since M is a ℤ_{n}-algebra, then Corollary 8 and Lemma 6 complete the proof. □

Corollary 10. For every nonnegative integer n which is not equal to 1 there exists a minimal non-totally minimal separable metrizable locally compact unital ring with char(R) = n.

Proof. Fix a natural number n ≥ 2. Let F_{i} = ℤ_{n} for every i ∈ ℕ. Consider the topological ring product ( ∏_{i∈ℕ} F_{i}, σ) and the dense countable topological subring ( ∑_{i∈ℕ} F_{i}, τ) . Denote by τ_{d} the discrete topology on R: = ∑_{i∈ℕ} F_{i}. Clearly, the ℤ_{n}-

unitalization R_{+} of (R, τ_{d}) is not a minimal ring because we can take on R_{+} the (strictly coarser) ring topology of the ℤ_{n}-unitalization for (R, τ). On the other hand, by Theorem 9, the discrete non-minimal ring R_{+} is a continuous ring retract of a minimal ring M_{+}. Eventually, M_{+} is the desired ring.

For the case n = 0, consider the ring product ℝ × M_{+}, where M_{+} is a minimal ring constructed for the case n ≥ 2, and use the productivity of the class of minimal unital rings [3].

Acknowledgement. I would like to express my gratitude to D.Dikranjan for helpful comments and suggestions.

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e-mail: megereli@bimacs.cs.biu.ac.il