@ Appl. Gen. Topol. 15, no. 2(2014), 229-233doi:10.4995/agt.2014.2230
c© AGT, UPV, 2014

Radicals in the class of compact right

topological rings

Mihail Ursul and Adela Tripe

a
Department of Mathematics and Computer Science, PNG University of Technology, Private

Mail Bag Lae 411, Lae, Papua New Guinea (mihail.ursul@gmail.com)
b
Department of Mathematics and Computer Science, University of Oradea, str. Universităţii No.

1, Oradea, Romania (tadela@uoradea.ro)

Abstract

We construct in this article three radicals in the class of compact right

topological rings. We also prove that a simple left Noetherian compact

right topological ring of prime characteristic is finite.

2010 MSC: Primary 16W80; Secondary 54HXX.

Keywords: right topological ring; radical; left Noetherian topological ring.

1. Introduction

The notion of a compact left (right) topological ring was introduced in [6]
and later in [3].

A fundamental question occuring at the study of any class of rings is: What
are simple rings which are in this class?

It is known, for instance, that a compact ring is simple if and only if it is a
matrix ring over a finite division ring [4].

We study in this paper simple compact right topological rings. We prove
that every left Noetherian compact right topological simple ring with identity of
prime characteristic is finite. Answering a question posed by Professor Richard
Wiegandt, we construct some radicals in the class of compact right topological
rings.

Received 24 March 2014 – Accepted 24 July 2014

http://dx.doi.org/10.4995/agt.2014.2230


M. Ursul and A. Tripe

2. Notation and conventions

All rings are assumed associative not necessarily with identity. Topological
groups are assumed Hausdorff. The closure of a subset A of a topological space
is denoted by A. A J-semismiple ring is a ring whose Jacobson radical is zero.

3. A few elementary results

Differences between elementary properties of topological rings and right
topological rings are essential. We will give an example of a right topolo-
gical ring for which the right annihilator is not closed. We construct also a
compact right topological ring with nonclosed center.

Let R be a compact ring with identity and α : R → R a group endo-
morphism. Define on the group A = R × R with the product topology the
multiplication (a, x)(b, y) = (0, aα(b)). Then A3 = 0 and A is a compact right
topological ring.

Let p be a prime number and R = Fℵ0p with the product topology or R =
Fp[[t]] be the ring of formal power series over Fp. Then the additive group
of R contains a dense subgroup B of index p. Let α : R → R be a group
homomorphism for which ker α = B. Then AnnrA ⊃ R × B, hence AnnrA is
dense. Since AnnrA 6= A, it is not closed.

Theorem 3.1. Let (A, T ) be a compact abelian group.Then the ring EndA of
all not necessarily continuous endomorphisms of A with the topology of point-
wise convergence is a compact right topological ring.

Proposition 3.2. For any right topological ring R the closure of its abstract
center Z = Z(R) is a subring.

Proof. Indeed, Z · Z ⊂ Z · Z = Z · Z ⊂ Z = Z. �

Example 3.3. A compact topological ring with nonclosed center.
The group R admits a compact group topology (see [2]). According to

Theorem 3.1 EndR admits a compact right ring topology. Since the center of
EndR is Q, the center is not closed. The subring Q of EndR is connected.

4. Simple compact right topological rings

As usual, a ring R is called simple if R2 6= 0 and R has no ideals different
from 0 and R. Recall that an ideal M of a ring R is called maximal if the
factor ring R/M is a simple ring.

Lemma 4.1. There are no infinite compact right topological simple Artinian

rings of prime characteristic.

Proof. Assume on the contrary that there exists an infinite compact right topo-
logical simple Artinian ring R of prime characteristic.

There exists an infinite cardinal α such that R (+) ∼=top Z (p)
α
. We identify

the additive group R (+) of R with Z (p)
α
. Consider the ring S = Endc (R (+))

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Radicals in the class of compact right topological rings

of continuous endomorphisms of R (+) with the compact-open topology. We
note that Endc (R (+)) has a fundamental system of neighborhoods of 0 con-
sisting of right ideals. Indeed, denote for each neighborhood V of 0R by T (V )
the right ideal {ρ ∈ S | ρ (R) ⊂ V }. The family {T (V )}, where V runs all
neighborhoods of R, forms a fundamental system of neighborhoods of 0S con-
sisting of right ideals. We shall estimate now the weight of S. Fix β < α . Put

Hβ =

{

ρ ∈ S | ρ (R) ⊂ Z (p)β ×
∏

γ 6=β

{0}γ

}

.

Fact 1. Hβ is a discrete right ideal of S.
Indeed, Hβ ∩ T ({0} ×

∏

γ 6=β Z(p)γ}) = 0.

Fact 2. |Hβ| ≤ α.
Indeed, fix β0, . . . , βn < α, n ∈ ω.
Put Sβ0...βn = {χ ∈ Hβ | χ({0}β0 × · · ·× {0}βn ×

∏

γ /∈{β0,...,βn}
Z(p)γ) = 0}.

Since the values of every π ∈ Sβ0...βn are determined by the set Z(p)β0 × · · · ×
Z(p)βn ×

∏

γ /∈{β0,...,βn}
{0}γ, Sβ0...βn is finite. This implies that |Hβ| ≤ α.

Fact 3. |
∑

β<α Hβ| ≤ α.

Since |Hβ| ≤ α for every β < α, the cardinality of
∑

β<α Hβ is ≤ α×α = α.

Fact 4.
∑

β<α Hβ is a dense subgroup of S.

Denote for each β < α the endomorphism qβ of R(+): qβ : R → R, (xγ) 7→
xβ ×

∏

δ 6=β 0δ and by prβ the projection of R on Z(p)β = Z(p). Evidently, qβ ∈
H. We have that qβ0ϕ + · · · + qβnϕ ∈ H for each β0, . . . , βn < α, n ∈ ω, ϕ ∈ S.

Set V = {0}β0 × · · · × {0}βn ×
∏

β 6=β0,...,βn
Z(p)β. Then we have:

prβi[(ϕ − Σ
n
j=0qβj ϕ)(x)] = prβi[ϕ(x) − Σ

n
j=0qβj ϕ(x)]

= prβi[ϕ(x)] − Σ
n
j=0prβi[qβj ϕ(x)]

= prβi[ϕ(x)] − prβi[ϕ(x)]
= 0 ,

for every i ∈ {0, n} and every x ∈ R.
We obtain (ϕ − Σnj=0qβj ϕ)(R) ⊂ V , i.e., ϕ − Σ

n
j=0qβj ϕ ∈ T (V ), hence

ϕ ∈ H + T (V ).
We have proved that S = H + T (V ). The topological group S has a fun-

damental system of neighborhoods of 0 of cardinality ≤ α. It follows that
w(S) ≤ α, where w(S) is the weight of S. Define χ : R → S, x 7→ Rx. Then
χ is a ring anti-isomorphism of R in S. Since S has a fundamental system of
neighborhoods of 0 consisting of right ideals, |χ(R)| ≤ α, a contradiction, since
|R| = 2α > α. �

Theorem 4.2. Every simple left Noetherian compact right topological ring R
with identity of prime characteristic is finite.

Proof. Since R is a compact right topological ring, every principal left ideal
of R is closed. By compactness, every finitely generated left ideal is closed.
Since R is left Noetherian, every left ideal of R is closed. If F is a filter basis
consisting of cosets with respect to left ideals, then by compactness of R its
intersection is non-empty. Thus, R is a discrete left linearly compact simple

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M. Ursul and A. Tripe

ring. It follows that R is a left linearly compact discrete simple ring. Then,
(see [7], [5]), R is an Artinian simple ring. By Lemma 4.1, R is finite. �

5. Radicals in the class of compact right topological rings

Lemma 5.1. If R is a compact right topological ring and I an ideal with
identity e, then I is a topological direct summand.

Proof. It is well-known that e will be a central idempotent. We have that Re
and R(1−e) = {x ∈ R | xe = 0} are closed ideals of R. Since R = Re⊕R(1−e)
is a direct sum, by compactness of R, this sum is topological. �

Let L be a class of Hausdorff right topological rings. We will say that the
operator ρ is a radical in the class L if:

(i) ρ(R) is a closed ideal of R for each R ∈ L;
(ii) ρ(ρ(R)) = ρ(R);
(iii) f(ρ(R)) ⊂ ρ(f(R)) for any continuous homomorphism f : R → S, where

R, S ∈ L;
(iv) ρ(R/ρ(R)) = 0.

Remark 5.2. Our definition differs from the definition of a radical in classes
of topological rings introduced in [1]. In [1] it is assumed that the class of
topological rings for which is defined a radical is closed under taking of ideals.
Evidently, this condition is not filled for compact right topological rings.

We will construct three radicals in the class of compact right topological
rings. First, we note that the connected component of zero is a radical in
the class of compact right topological rings. Indeed, if R is a compact right
topological ring, then its component R0 of zero is equal to ∩n∈NnR, hence R0
is a closed ideal of R.

Theorem 5.3. Let L be the class of all compact right topological rings and
ρ(R) = ∩M, where M runs all open maximal ideals of R. Then ρ is a radical.

Proof. Since every open ideal is closed, ρ(R) is a closed ideal of R. Then
ρ(R/ρ(R) = 0 and by Kaplansky’s Theorem R/ρ(R) has identity. The factor
ring R/ρ(R) is a compact J-semisimple ring.

Denote by H the two-sided ideal generated by ρ(ρ(R)). Then ρ(ρ(R)) ⊂
H ⊂ ρ(R). By Andrunakievich’s Lemma, H3 ⊂ ρ(ρ(R)). Since ρ(R)/ρ(ρ(R))
is semiprime, H ⊂ ρ(ρ(R)), hence H = ρ(ρ(R)).

We claim that if f : R → S is a continuous homomorphism, then f(ρ(R)) ⊂
ρ(f(R)).

Indeed, consider f as a surjective homomorphism of R on L = f(R). Then
ρ(L) = ∩M′ where M′ runs all maximal ideals of L. It follows that f−1(ρ(L)) =
∩f−1(M′) where M′ runs all open maximal ideals of L. If M′ is a maximal
open ideal of L, then f−1(M′) will be an open maximal ideal of R. It follows
that f−1(ρ(L)) ⊃ ρ(R), hence f(ρ(R)) ⊂ ρ(f(L)).

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Radicals in the class of compact right topological rings

Now we will prove that ρ2 = ρ. Indeed, consider the factor ring R/ρ(ρ(R)).
The ideal ρ(R)/ρ(ρ(R)) has identity, therefore is a topological direct summand.
We obtain that R/ρ(ρ(R)) is a topological direct sum of R/ρ(R) and ρ(ρ(R)).
It follows that ρ(R/ρ(ρ(R))) = 0, hence ρ(R) ⊂ ρ(ρ(R)) and so ρ(ρ(R)) =
ρ(R). �

We will obtain another radical φ different from ρ if we will put φ(R) the
intersection of all open ideals V of a compact right topological ring R for
which R/V is a field.

Theorem 5.4. For any compact right topological ring R and for any its closed
ideal I holds ρ(I) = I ∩ ρ(R).

Proof. If φ : R → R/ρ(R) is the natural homomorphism, then the restriction
φ |I: I → I + ρ(R)/ρ(R) is surjective and I + ρ(R)/ρ(R) is a closed ideal of
R/ρ(R). Since R/ρ(R) is a product of finite simple rings ([4]), I + ρ(R)/ρ(R)
is ρ-semisimple. Thus ρ(I) ⊂ ρ(R) ∩ I.

Denote by H the two-sided ideal of R generated by ρ(I). Then ρ(R) ⊂
H ⊂ I. By Andrunakievich’s Lemma, H3 ⊂ ρ(R). Since I/ρ(I) is semiprime,
H ⊂ ρ(I), hence H = ρ(I).

Since I/ρ(I) is a compact semisimple ring, it has an identity [4]. By Lemma
5.1 there exists a closed ideal S1 of R/ρ(R) such that R = S1 ⊕ (I/ρ(I)), a
topological direct sum. By a standard result, S1 = S/ρ(R), where S is a closed
ideal of R. We have that R = S + I and S ∩ I = ρ(I).

The factor ring R/S is topologically isomorphic to I/I ∩ S = I/ρ(I), hence
is ρ-semisimple. It follows that ρ(R) ⊂ S , hence ρ(R) ∩ I ⊂ S ∩ I = ρ(I). �

Acknowledgements. The authors are grateful to Professor Richard Wie-
gandt for his interest in the results of this article. We are also indebted to
Professor Fred van Oystaeyen whose remarks stimulated the proof of Theorem
4.2. We are also grateful to the refree for her/his important sugestions.

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