Morita equivalent unital locally matrix algebras
We describe Morita equivalence of unital locally matrix algebras in terms of their Steinitz parametrization. Two countable-dimensional unital locally matrix algebras are Morita equivalent if and only if their Steinitz numbers are rationally connected. For an arbitrary uncountable dimension α and an...
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Bezushchak, O. Oliynyk, B. 2023-03-03T19:33:11Z 2023-03-03T19:33:11Z 2020 Morita equivalent unital locally matrix algebras / O. Bezushchak, B. Oliynyk // Algebra and Discrete Mathematics. — 2020. — Vol. 29, № 2. — С. 173–179. — Бібліогр.: 18 назв. — англ. 1726-3255 2010 MSC: 03C05, 03C60. DOI:10.12958/adm1545 https://nasplib.isofts.kiev.ua/handle/123456789/188513 We describe Morita equivalence of unital locally matrix algebras in terms of their Steinitz parametrization. Two countable-dimensional unital locally matrix algebras are Morita equivalent if and only if their Steinitz numbers are rationally connected. For an arbitrary uncountable dimension α and an arbitrary not locally finite Steinitz number s there exist unital locally matrix algebras A, B such that dimF A = dimF B = α, st(A) = st(B) = s, however, the algebras A, B are not Morita equivalent. The second author was partially supported by the grant for scientific researchers of the “Povir u sebe” Ukranian Foundation. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Morita equivalent unital locally matrix algebras Article published earlier |
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Morita equivalent unital locally matrix algebras |
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Morita equivalent unital locally matrix algebras Bezushchak, O. Oliynyk, B. |
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Morita equivalent unital locally matrix algebras |
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Morita equivalent unital locally matrix algebras |
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Morita equivalent unital locally matrix algebras |
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Morita equivalent unital locally matrix algebras |
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morita equivalent unital locally matrix algebras |
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Bezushchak, O. Oliynyk, B. |
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Bezushchak, O. Oliynyk, B. |
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We describe Morita equivalence of unital locally matrix algebras in terms of their Steinitz parametrization. Two countable-dimensional unital locally matrix algebras are Morita equivalent if and only if their Steinitz numbers are rationally connected. For an arbitrary uncountable dimension α and an arbitrary not locally finite Steinitz number s there exist unital locally matrix algebras A, B such that dimF A = dimF B = α, st(A) = st(B) = s, however, the algebras A, B are not Morita equivalent.
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Morita equivalent unital locally matrix algebras / O. Bezushchak, B. Oliynyk // Algebra and Discrete Mathematics. — 2020. — Vol. 29, № 2. — С. 173–179. — Бібліогр.: 18 назв. — англ. |
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AT bezushchako moritaequivalentunitallocallymatrixalgebras AT oliynykb moritaequivalentunitallocallymatrixalgebras |
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2025-11-26T01:45:40Z |
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2025-11-26T01:45:40Z |
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1850606146468970496 |
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“adm-n2” — 2020/7/8 — 8:15 — page 173 — #33
© Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 29 (2020). Number 2, pp. 173–179
DOI:10.12958/adm1545
Morita equivalent unital locally matrix algebras∗
O. Bezushchak and B. Oliynyk
Communicated by A. P. Petravchuk
Abstract. We describe Morita equivalence of unital locally
matrix algebras in terms of their Steinitz parametrization. Two
countable-dimensional unital locally matrix algebras are Morita
equivalent if and only if their Steinitz numbers are rationally con-
nected. For an arbitrary uncountable dimension α and an arbitrary
not locally finite Steinitz number s there exist unital locally matrix
algebras A, B such that dimF A = dimF B = α, st(A) = st(B) = s,
however, the algebras A, B are not Morita equivalent.
Introduction
Let F be a ground field. Throughout the paper we consider unital
associative F–algebras. An algebra A with a unit 1A is called a unital
locally matrix algebra if an arbitrary finite collection of elements a1, . . .,
as ∈ A lies in a subalgebra B, 1A ∈ B ⊂ A, that is isomorphic to a matrix
algebra Mn(F ), n > 1.
The idea of parametrization of unital locally matrix algebras with
Steinitz numbers was introduced by J. G. Glimm [1]. Diagonal locally
simple Lie algebras of countable dimension were parametrized with Steinitz
numbers by A. A. Baranov and A. G. Zhilinskii in [2, 3]. The extension of
these results to regular relation structures was done in [4].
In this paper we apply Steinitz parametrisation to Morita equivalence
classes of unital locally matrix algebras. We show that two countable-
dimensional unital locally matrix algebras are Morita equivalent if and
∗The second author was partially supported by the grant for scientific researchers
of the “Povir u sebe” Ukranian Foundation.
2010 MSC: 03C05, 03C60.
Key words and phrases: locally matrix algebra, Steinitz number, Morita equiv-
alence.
https://doi.org/10.12958/adm1545
“adm-n2” — 2020/7/8 — 8:15 — page 174 — #34
174 Morita equivalent unital locally matrix algebras
only if their Steinitz numbers are rationally connected. This result does not
extend to the uncountable case. Moreover, for an arbitrary uncountable
dimension α and an arbitrary not locally finite Steinitz number s there
exist unital locally matrix algebras A, B such that dimF A = dimF B = α,
st(A) = st(B) = s, however, the algebras A, B are not Morita equivalent.
1. Preliminaries
Let P be the set of all primes and N be the set of all positive integers.
A Steinitz number (see [5]) is an infinite formal product of the form
∏
p∈P
prp , (1)
where rp ∈ N∪ {0,∞} for all p ∈ P. The product of two Steinitz numbers
∏
p∈P
prp and
∏
p∈P
pkp
is a Steinitz number ∏
p∈P
prp+kp ,
where we assume, that t+∞ = ∞+ t = ∞+∞ = ∞ for all non-negative
integers t.
Denote by SN the set of all Steinitz numbers. Note, that the set N is
a subset of SN.
A Steinitz number (1) is called locally finite if rp 6= ∞ for any p ∈ P.
The numbers SN \ N are called infinite Steinitz numbers.
J. G. Glimm [1] parametrised countable-dimensional locally matrix
algebras with Steinitz numbers. In [6] we studied Steinitz numbers of
unital locally matrix algebras of arbitrary dimensions.
Let A be an infinite-dimensional locally matrix algebra with a unit 1A
over a field F and let D(A) be the set of all positive integers n such that
there is a subalgebra A′, 1A ∈ A′ ⊆ A, A′ ∼= Mn(F ).
Definition 1. The least common multiple of the set D(A) is called the
Steinitz number st(A) of the algebra A.
Given two unital locally matrix algebras A and B their tensor product
A⊗F B is a unital locally matrix algebra and st(A⊗F B) = st(A) · st(B)
(see [7]). In particular, a matrix algebra Mk(A) is a unital locally matrix
algebra and st(Mk(A)) = k · st(A).
“adm-n2” — 2020/7/8 — 8:15 — page 175 — #35
O. Bezushchak, B. Oliynyk 175
Theorem 1 ([1], see also [4]). If A and B are unital locally matrix
algebras of countable dimension then A and B are isomorphic if and
only if st(A) = st(B).
Let A be an algebraic system. The universal elementary theory UTh(A)
consists of universal closed formulas (see [8]) that are valid on A. The
systems A and B of the same signature are universally equivalent if
UTh(A) = UTh(B).
In [6] we showed that for unital locally matrix algebras A, B of di-
mension > ℵ0 the equality st(A) = st(B) does not necessarily imply that
A and B are isomorphic. However, st(A) = st(B) is equivalent to A, B
being universally equivalent.
2. Morita equivalence
Definition 2. Two unital algebras A, B are called Morita equivalent if
categories of their left modules are equivalent.
Let e ∈ A be an idempotent. We refer to the subalgebra eAe as a
corner of the algebra A. An idempotent e ∈ A is said to be full if AeA = A.
K.Morita [9] (see also [10, 11]) proved that the algebras A, B are Morita
equivalent if and only if there exists n > 1 and a full idempotent e in the
matrix algebra Mn(A) such that B ∼= eMn(A)e. Thus B is isomorphic to
a corner of the algebra Mn(A).
We say that a property P is Morita invariant if any two Morita
equivalent algebras do satisfy or do not satisfy P simultaneously.
An F -algebra A is a tensor product of finite-dimensional matrix alge-
bras if
A ∼= ⊗i∈IAi, Ai
∼= Mni
(F ), ni > 1.
Every tensor product (see [11]) of finite-dimensional matrix algebras is
a locally matrix algebra. G. Köthe [12] showed that the reverse is true
for countable-dimensional algebras. A.G.Kurosh [13] (see also [7, 14])
constructed examples of locally matrix algebras that do not decompose
into a tensor product of finite-dimensional matrix algebras.
Lemma 1. (1) Being a locally matrix algebra is a Morita invariant
property.
(2) Being a tensor product of finite-dimensional matrix algebras is a
Morita invariant property.
Proof. (1) Let algebras A, B be Morita equivalent. Then there exists
n > 1 and a full idempotent e ∈ Mn(A) such that B ∼= eMn(A)e. If the
algebra A is locally matrix then so is the matrix algebra Mn(A). J.Dixmier
“adm-n2” — 2020/7/8 — 8:15 — page 176 — #36
176 Morita equivalent unital locally matrix algebras
[15] showed that a corner of a locally matrix algebra is a locally matrix
algebra. Hence B is a locally matrix algebra.
(2) Now suppose that A ∼= ⊗i∈IAi, Ai
∼= Mni
(F ), ni > 1. Then
Mn(A) ∼= Mn(F )⊗F A ∼= Mn(F )⊗F (⊗i∈IAi).
There exists a finite subset I0 ⊂ I, | I0 |< ∞, such that e ∈ Mn(F ) ⊗F
(⊗i∈I0Ai). As above, the corner e(Mn(F ) ⊗F (⊗i∈I0Ai))e is a matrix
algebra. Hence
B ∼= eMn(A)e ∼= e(Mn(F )⊗F (⊗i∈I0Ai))e⊗F (⊗i∈I\I0Ai),
which completes the proof of the lemma.
Definition 3. We say that nonzero Steinitz numbers s1, s2 are rationally
connected if there exists a rational number q ∈ Q such that s2 = q · s1.
Theorem 2. 1) If unital locally matrix algebras A, B are Morita equivalent
then their Steinitz numbers st(A), st(B) are rationally connected.
2) If unital locally matrix algebras A, B are countable-dimensional
then they are Morita equivalent if and only if st(A), st(B) are rationally
connected.
3) For an arbitrary not locally finite Steinitz number s there exist
not Morita equivalent unital locally matrix algebras A, B of arbitrary
uncountable dimensions such that st(A) = st(B) = s.
4) For a countable-dimensional unital locally matrix algebra A the
Morita equivalence class of A is countable up to isomorphism. For a unital
locally matrix algebra of an arbitrary dimension the Morita equivalence
class is countable up to universal equivalence.
Remark 1. Countability of Morita equivalence classes of finitely presented
algebras was discussed in [16–18].
Let A be a locally matrix algebra, let a ∈ A. There exists a subalgebra
1A ∈ A1 < A, a ∈ A1, such that A1
∼= Mn(F ), n > 1. Let r be the range
of the matrix a in A1. Let
r(a) =
r
n
, 0 6 r(a) 6 1.
V.M.Kurochkin [14] noticed that the number r(a) does not depend on a
choice of the subalgebra A1. We will call r(a) the relative range of the
element a.
Lemma 2. Let e be an idempotent of a locally matrix algebra A. Then
st(eAe) = r(e) · st(A).
“adm-n2” — 2020/7/8 — 8:15 — page 177 — #37
O. Bezushchak, B. Oliynyk 177
Proof. Consider the family of all matrix subalgebras 1A ∈ Ai < A, Ai
∼=
Mni
(F ), i ∈ I, such that e ∈ Ai. Then st(A) = lcm(ni, i ∈ I). The range
of the matrix e in Ai is equal to r(e) · ni. Hence
eAie ∼= Mr(e)·ni
(F ) and st(eAe) = lcm(r(e) · ni, i ∈ I) = r(e) · st(A).
Proof of Theorem 2. 1) Let A, B be locally matrix algebras that are
Morita equivalent. Hence there exists k > 1 and an idempotent e ∈ Mk(A)
such that B ∼= eMk(A)e. Let r(e) be the relative range of the idempotent
e in the locally matrix algebra Mk(A). By Lemma 2
st(B) = r(e) · st(Mk(A)) = r(e) · k · st(A).
Since the number r(e) · k is rational it follows that the Steinitz numbers
st(A), st(B) are rationally connected.
2) Let A, B be countable-dimensional locally matrix algebras. Suppose
that their Steinitz numbers st(A), st(B) are rationally connected. Our
aim is to prove that the algebras A, B are Morita equivalent. There exist
integers k, l > 1 such that k · st(A) = l · st(B). Consider the matrix
algebras Mk(A) and Ml(B). We have
st(Mk(A)) = k · st(A) = l · st(B) = st(Ml(B)).
By Glimm’s Theorem [1] the algebras Mk(A) and Ml(B) are isomorphic.
Hence the algebras A, B are Morita equivalent.
3) Let S be a not locally finite Steinitz number. In [7] (see also [6]
and [13]) we showed that there exists a locally matrix algebra A of an
arbitrary uncountable dimension α such that st(A) = s and A is not
isomorphic to a tensor product of finite dimensional matrix algebras. It
is easy to see that there exists a locally matrix algebra B of dimension
α such that st(B) = s and B is isomorphic to a tensor product of finite-
dimensional matrix algebras. By Lemma 1 (2) the algebras A, B are not
Morita equivalent.
4) For a countable-dimensional locally simple algebra A all algebras
in its Morita equivalence class have Steinitz numbers q · st(A), where q is
a positive rational number, and are uniquely determined by their Steinitz
numbers up to isomorphism. This implies that the Morita equivalence
class of A is countable.
If the algebra A is not necessarily countable-dimensional then Steinitz
numbers q · st(A) determine universal elementary theories of algebras in
this class (see [6]). Hence the Morita equivalence class of A is countable
up to universal equivalence. This completes the proof of Theorem 2.
“adm-n2” — 2020/7/8 — 8:15 — page 178 — #38
178 Morita equivalent unital locally matrix algebras
If nonzero Steinitz numbers s1, s2 are rationally connected then it
makes sense to talk about their ratio q = s2
s1
which is a rational number.
For a countable-dimensional locally matrix algebra A its Morita equiva-
lence class is ordered: for algebras A1, A2 in this class we say that A1 < A2
if
st(A1)
st(A2)
< 1.
Proposition 1. Let A1, A2 be countable-dimensional Morita equivalent
locally matrix algebras. Then
st(A1)
st(A2)
< 1 if and only if A1 is isomorphic to a proper corner of A2.
Proof. If A1
∼= eA2e, where e is a proper idempotent of the algebra A2,
then st(A1) = r(e)st(A2) by Lemma 2. Hence
st(A1)
st(A2)
= r(e) < 1.
Now let
st(A1)
st(A2)
=
m
n
< 1,
where m, n are relatively prime integers. Then n is a divisor of st(A2).
Hence the algebra A2 contains a subalgebra 1 ∈ A
′
2 < A2, A
′
2
∼= Mn(F ).
Hence (see [13])
A2
∼= A
′
2 ⊗F C ∼= Mn(C),
where C is the centralizer of the subalgebra A
′
2 in A2. Consider the
idempotent e = diag(1, 1, . . . , 1
︸ ︷︷ ︸
m
, 0, . . . , 0) ∈ Mn(C). By Lemma 2
st(eMn(C)e) =
m
n
st(A2) = st(A1).
By Glimm’s Theorem A1 is isomorphic to a corner of Mn(C), hence to a
corner of A2.
References
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Comm. Algebra, 27, no. 6 (1999), 2749–2766.
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Contact information
Oksana Bezushchak Faculty of Mechanics and Mathematics, Taras
Shevchenko National University of Kyiv,
Volodymyrska, 60, Kyiv 01033, Ukraine
E-Mail(s): bezusch@univ.kiev.ua
Bogdana Oliynyk Department of Mathematics, National
University of Kyiv-Mohyla Academy, Skovorody
St. 2, Kyiv, 04070, Ukraine
E-Mail(s): oliynyk@ukma.edu.ua
Received by the editors: 09.02.2020.
O. Bezushchak, B. Oliynyk
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