1. Introduction
The symmetric ring of quotients
of a semiprime ring
is probably the most comfortable ring of quotients of
. This notion was first introduced by W.S. Martindale [1] for prime rings and extended to the semiprime case by Amitsur [2] . Recall that a ring
is said to be semiprime (resp. prime) if
for every nonzero ideal
of
(resp. if
for all nonzero ideals
of
). The center
of
is called the extended centroid of
, and the
-subring
of
generated by
is called the central closure of
. A semiprime
is said to be centrally closed whenever
. For every
, we will denote
and
the left and right multiplication operators, respectively, by
on
. The multiplication ring of
,
, is defined as the subring of
generated by the identity operator
and the set
. The goal of this paper is to give a semiprime extension of the following well-known result (see for instance [3] , Theorem A.9):
“If the multiplication ring of a centrally closed prime ring
has a finite rank operator over
then
contains an idempotent
such that
is a division algebra finitely generated over
”.
It is also well know that the extended centroid of a prime ring is a field, however, for a semiprime ring, we can only assert that said extended centroid is a von Neumann regular ring. This is the cause of the difficulty of extending this result. The starting point of this path relies on the fact that each subset
of
has an associated idempotent
of the extended centroid
(see [4] , Theorem 2.3.9) and on a consequence (see [4] , Theorem 2.3.3 and Proposition 1.1 below) of the Weak Density Theorem ([4] , Theorem 1.1.5).
2. Tools
We shall assume throughout this paper that
is a centrally closed semiprime ring. First of all, we recall that if
is the set of all idempotents in
has a partial order given by
iff
. Moreover,
is a Boolean algebra for the operations
![]()
In fact, [5] , Theorem 1.8 remains valid in case that
is a ring, and so this Boolean algebra is complete, that is, every subset of
admits supremum and infimum. We will use the properties of the idempotent associated to a subset referred to in ([4] , Theorem 2.3.9 (i) and (ii)) without notice.
Given a
-submodule
of
, we will say that
is
-finitely generated if there exist
such that
.
Next, we establish our main tool.
Proposition 1.1 Let
be a
-finitely generated
-submodule of
, and let
. Then there exists
such that: a)
, b)
and c)
.
Proof. We denote
. If
, then
. Suppose that
. If
, then we take
. In other case, take
, for some
. By ([4] Theorem 2.3.9), there exists
such that
and
. In particular,
, and
. Thus, the family
of all nonzero idempotents satisfying
and
is not empty. Let
. Note that
because of completeness of
, and, of course,
. If
, then, by ([4] , Theorem 2.3.3), there exists ![]()
such that
and
. But, since
, we have
and so ![]()
for all
. Hence
, that is,
, which is a contradiction with
. Therefore
belongs to
. Take
. Let us see that
. Indeed, for every
, we can write:
(1)
Moreover, if there exists
and
such that
![]()
then
. Take
such that
and
is an idempotent in
. It is clear that
, and so
by maximality. Thus,
and
. Finally, note that:
![]()
Thus, the sum is direct. Note that
verifies properties a), b) and c). ![]()
As a consequence, we have the following:
Corollary 1.2 Let
be a nonzero C-submodule of
and
such that
. Then there exists
such that
.
Proof. If
take
. In other case,
. By Proposition 1.1, there is
such that
and
. Thus,
, and so,
. ![]()
Note that if
then it may be that
but
. This forces us to make a convenient definition of set
-linearly independent. We will say that
nonzero elements
of R are C-linearly independent (or that the set
is
-linearly independent) if, for all
,
implies
for all
, or equivalently, if the
-linear envelope
of the subset
S satisfies:
. Note that for every
and
, if
and
are nonzero, then
the sets
and
are C-linearly independent and both generate the C-module
. In general, any C-finitely generated C-module
can be obtained as the C-linear envelope of C-linearly independent sets with different cardinal. In this sense, in ([4] Theorem 2.3.9. (iv)) is asserted that one can select a C-linearly independent set with a minimal number of generators under certain conditions. In any case, certain properties of the vector spaces remain true for the C-submodules: the next results, probably well-known, are obtained as a consequence of Proposition 1.1.
Corollary 1.3 Let
be a subset of
and
two C-finitely generated C-submodules of
such that
. Then there are
such that the subset of ![]()
![]()
is
-linearly independent, and
.
Proof. If
, we take
. In other case, by Proposition 1.1, there exists
such that
. Now, if
then take
, and if
then,
by Proposition 1.1, there exists
such that
. To conclude, it is enough to repeat this procedure
times. ![]()
Corollary 1.4 If
is a C-finitely generated C-submodule then there exist
and ![]()
such that
.
Proof. Let
such that
. By Corollary 1.3 we can assume that the set
is C-linearly independent.
It is clear that
. By Proposition 1.1, there exist
such that, for every
,
and
![]()
Hence,
![]()
Therefore,
Analogously, since
with
and
, we have
![]()
and so,
.
By repeating this procedure, there are
such that
![]()
and hence,
. Therefore, since,
with
and
, and, for each
,
with
and
, we deduce that
![]()
and so,
. Again, by Corollary 1.3, we obtain
-linear independent
elements of
such that
. ![]()
Let
be a right ideal of R. We say that
is a
-minimal right ideal if for every nonzero right ideal
of
contained in
, there exists some
such that
. Note that if
is prime then, since
is a field,
, and so, the concepts of
-minimal right ideal and minimal right ideal agree.
Recall that for a subset
of
the left annihilator
will be denoted by
. The right annihilator
is similarly defined.
Proposition 1.5 Let
be a
-minimal right ideal of
. Then there exists an idempotent
and
such that
. As a consequence
is a
-minimal ideal of
.
Proof. Since
and R is semiprime,
, and hence there exists
such that
. Note that this implies the existence of some
such that
. Since
, there exists
such that
. Note that
, and then:
, that is,
. Since
is a right ideal of
, if
, by minimality there exists
such that
. But, since
, we have
, a contradiction. Hence,
(
because
). Then
. Since
is
-minimal, there exists some
such that
. ![]()
We finalized this section with a desirable result, which is similar to the well-known result for minimal right ideals (see for instance [4] , Proposition 4.3.3).
Proposition 1.6 Let
be an idempotent of
. The following assertion are equivalent:
1)
is
-minimal right ideal of
.
2) For every
there exist
and
such that
.
Proof. (1)
(2). Since
is an idempotent, it is clear that
is the unit of
. Take
. It is clear that
, and so, since xR is right ideal of R, there exists
such that
. In particular, there is
such that
. Therefore
.
(2)
(1)
Let I be a nonzero right ideal of R such that
. Let us see that there exists
such that
. Indeed, if we take
, by semiprimeness of R, there exists
such that
. Note that
for every
. Consequently,
is a nonzero element of
, and hence there are
and
such that
. Therefore
, and so,
. Thus
. ![]()
A nonzero idempotent q of R is said to be
-minimal when the above assertions are fulfilled.
3. Theorem
In this section we will prove a semiprime extension of [3] , Theorem A.9. Concretely,
Theorem 2.1 Let R be a centrally closed semiprime ring. Then
has a C-finite rank operator if, and only if,
contains a
-minimal idempotent
such that
is
-finitely generated.
We begin this proof with an another consequence of Proposition 1.1,which is an improvement of Corollary 1.2 to case
. Given a nonzero C-module M C-finitely generated, we will say that
when- ever
![]()
Lemma 2.2 Let
be a nonzero
-submodule of
and suppose that, for every
such that
,
. If
for some
then there exists
such that
.
Proof. It is clear that
. By Proposition 1.1, there exist
such that
![]()
and
, in fact,
. Moreover,
![]()
Hence,
![]()
If
, then
![]()
that is,
, and this is a contradiction. Thus,
and
![]()
Note that if
then
, which is a contradiction. By Proposition 1.1, there exist
such that
![]()
and
. Therefore, since
with
and
, it is clear that
![]()
Hence,
![]()
If
, then
is contained in
summands, which is a contradiction. Hence, since
, we have
![]()
Note that if
, then
, which is a contradiction. By repeating this procedure, we find
such that,
,
, and
.
Therefore, denoting
, again by Proposition 1.1, there exists
such that
and,
![]()
and hence,
,
or even
.
Of course,
because
, and so,
. Thus, take
. ![]()
The next result is an immediate consequence of the Weak Density (see [4] , Theorem 2.3.3). We will denote by
the operator
for all
.
Lemma 2.3 Let
. Assume that
or
are C-linearly inde- pendent sets such that
. Then there are
and
such that
.
Proof. Assume that
are C-linearly independent. If
for all
then,
since
, we deduce that
, is a contradiction. For simplicity, we
can suppose that
. By [4] (Theorem 2.3.3), there exists
with
, such that
and
for all
. Put
, and note that, for every
, we have:
.
As a consequence:
. Moreover, by [4] (Corollary 2.3.10),
. ![]()
First step in the proof of Theorem
Proposition 2.4 If
has a
-finite rank operator then there are
such that
is
- finitely generated.
Proof. First of all, given a nonzero operator
with C-finite rank we can find an operator of the
form
, which has also C-finite rank. In fact, the most general form of G is: ![]()
for some
, and
. We can take an element
such that
, because in other case we would have
, a contradiction. Analogously, there exists some
such that
. Now,
is a nonzero operator with the desired form. Moreover, if
is
-finitely generated then
is also
-finitely generated. Secondly, taking in mind Corollary 1.3, we can assume without loss of generality that the set
is C-linearly independent. Finally, by Lemma 2.3 there are ![]()
and
such that
, and so,
is also
-finitely generated. ![]()
Second step in the proof of Theorem is a consequence of Lemma 2.2, and its proof can be obtained from a careful reading of the proof of [4] (Lemma 6.1.4).
Proposition 2.5 Let
such that
is
-finitely generated. Then there exist a
-minimal idempotent
such that
is
-finitely generated.
Proof. Without loss of generality we can assume that
. Since, in other case, if we take ![]()
then
. Suppose further that
, for
. By Corollary 1.3, we can assume that
the sum is direct. Consider the set
.
It is clear that
. Take m as the minimum of H and
such that
for some
. Let
. If
, then
, which is a contradiction because of semiprimeness of
. Thus
. Let
be a right ideal of
and
, where
. Setting
we note that
. Note that if
then
, a contradiction with the semi-
Primeness. Take
, it is clear that
. Note that
satisfies the hypothesis either of the Corollary 1.2 (if
) or of the Proposition 2.2 (if
), in any case, there is
such that
. In particular,
. Therefore,
, that is,
is a
-minimal right ideal of
. By Proposition 1.5, there exist
, and
such that
. Clearly
, and so
where
. Hence
and
so
is
-finitely generated. ![]()
Finally, the converse is obvious.
Funding
Supported by the Junta de Andaluca Grant FQM290.