﻿ On Finite Rank Operators on Centrally Closed Semiprime Rings

Vol.04 No.09(2014), Article ID:49454,6 pages
10.4236/apm.2014.49056

On Finite Rank Operators on Centrally Closed Semiprime Rings

J. C. Cabello1, R. Casas2, P. Montiel3   Received 2 July 2014; revised 2 August 2014; accepted 16 August 2014

ABSTRACT

We prove that the multiplication ring of a centrally closed semiprime ring R has a finite rank operator over the extended centroid C iff R contains an idempotent q such that qRq is finitely generated over C and, for each , there exist and e an idempotent of C such that .

Keywords:

Ring, Semiprime Ring, Extended Centroid, Minimal Idempotent 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  for prime rings and extended to the semiprime case by Amitsur  . 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  , 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  , Theorem 2.3.9) and on a consequence (see  , Theorem 2.3.3 and Proposition 1.1 below) of the Weak Density Theorem ( , 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,  , 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 ( , 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 ( 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 ( , 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 ( 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  , 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  , 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  , 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  (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  (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  (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.

References

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