Distinguising between idempotents of infinite matrices
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I am exploring the non-unital algebra $M_infty(K)$, which consists of all matrices over a field $K$ whose rows and oclumns are indexed by $Z^+ times Z^+$ which have only finitely many nonzero entries.
I am trying to distinguish between two idempotents, the first are idempotents of the form $e_{ii}$ where $e_{ij}$ is the matrix unit with 1 in the $(i,j)$ position and zeroes elsewhere. The second family of idempotents are of the form $$sum_{i = 1}^n e_{i1} quad text{ or} quad sum_{j = 1}^m e_{1j},$$ that is, idempotents which are some sum of matrix units from the first column or row.
According to Lam's First Course, Exercise 12.2*, both of these idempotents are primitive since, when we consider the elements of $M_infty(K)$ as endomorphisms of some vector space $V$, their image is one-dimensional.
These idempotents are clearly distinct in their function (for example, the former family additively generates a set of local units for $M_infty(K)$ while the latter does not. However, I cannot (up until now) find a ring-theoretic property (such as primitivity etc) which distinguishes between them.
Am I mistaken in my reading of Exercise 12.2*?
ring-theory soft-question
asked Dec 21 '18 at 2:09
dpb492dpb492
1116
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add a comment |
$begingroup$
I am exploring the non-unital algebra $M_infty(K)$, which consists of all matrices over a field $K$ whose rows and oclumns are indexed by $Z^+ times Z^+$ which have only finitely many nonzero entries.
I am trying to distinguish between two idempotents, the first are idempotents of the form $e_{ii}$ where $e_{ij}$ is the matrix unit with 1 in the $(i,j)$ position and zeroes elsewhere. The second family of idempotents are of the form $$sum_{i = 1}^n e_{i1} quad text{ or} quad sum_{j = 1}^m e_{1j},$$ that is, idempotents which are some sum of matrix units from the first column or row.
According to Lam's First Course, Exercise 12.2*, both of these idempotents are primitive since, when we consider the elements of $M_infty(K)$ as endomorphisms of some vector space $V$, their image is one-dimensional.
These idempotents are clearly distinct in their function (for example, the former family additively generates a set of local units for $M_infty(K)$ while the latter does not. However, I cannot (up until now) find a ring-theoretic property (such as primitivity etc) which distinguishes between them.
Am I mistaken in my reading of Exercise 12.2*?
ring-theory soft-question
asked Dec 21 '18 at 2:09
dpb492dpb492
1116
$endgroup$
1
$begingroup$
Additively generating a set of local units is a ring-theoretic property. What exactly are you looking for?
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– Qiaochu Yuan
Dec 21 '18 at 23:11
$begingroup$
You know what, I had a brain fart and forgot that. I think I had gotten my mind set on primitive idempotents and had some tunnel-vision.
$endgroup$
– dpb492
Jan 3 at 1:26
add a comment |
$begingroup$
I am exploring the non-unital algebra $M_infty(K)$, which consists of all matrices over a field $K$ whose rows and oclumns are indexed by $Z^+ times Z^+$ which have only finitely many nonzero entries.
I am trying to distinguish between two idempotents, the first are idempotents of the form $e_{ii}$ where $e_{ij}$ is the matrix unit with 1 in the $(i,j)$ position and zeroes elsewhere. The second family of idempotents are of the form $$sum_{i = 1}^n e_{i1} quad text{ or} quad sum_{j = 1}^m e_{1j},$$ that is, idempotents which are some sum of matrix units from the first column or row.
According to Lam's First Course, Exercise 12.2*, both of these idempotents are primitive since, when we consider the elements of $M_infty(K)$ as endomorphisms of some vector space $V$, their image is one-dimensional.
These idempotents are clearly distinct in their function (for example, the former family additively generates a set of local units for $M_infty(K)$ while the latter does not. However, I cannot (up until now) find a ring-theoretic property (such as primitivity etc) which distinguishes between them.
Am I mistaken in my reading of Exercise 12.2*?
ring-theory soft-question
asked Dec 21 '18 at 2:09
dpb492dpb492
1116
$endgroup$
I am exploring the non-unital algebra $M_infty(K)$, which consists of all matrices over a field $K$ whose rows and oclumns are indexed by $Z^+ times Z^+$ which have only finitely many nonzero entries.
I am trying to distinguish between two idempotents, the first are idempotents of the form $e_{ii}$ where $e_{ij}$ is the matrix unit with 1 in the $(i,j)$ position and zeroes elsewhere. The second family of idempotents are of the form $$sum_{i = 1}^n e_{i1} quad text{ or} quad sum_{j = 1}^m e_{1j},$$ that is, idempotents which are some sum of matrix units from the first column or row.
According to Lam's First Course, Exercise 12.2*, both of these idempotents are primitive since, when we consider the elements of $M_infty(K)$ as endomorphisms of some vector space $V$, their image is one-dimensional.
These idempotents are clearly distinct in their function (for example, the former family additively generates a set of local units for $M_infty(K)$ while the latter does not. However, I cannot (up until now) find a ring-theoretic property (such as primitivity etc) which distinguishes between them.
Am I mistaken in my reading of Exercise 12.2*?
ring-theory soft-question
ring-theory soft-question
asked Dec 21 '18 at 2:09
dpb492dpb492
1116
asked Dec 21 '18 at 2:09
dpb492dpb492
1116
asked Dec 21 '18 at 2:09
dpb492dpb492
1116
asked Dec 21 '18 at 2:09
dpb492dpb492
1116
asked Dec 21 '18 at 2:09
dpb492dpb492
1116
1116
1
$begingroup$
Additively generating a set of local units is a ring-theoretic property. What exactly are you looking for?
$endgroup$
– Qiaochu Yuan
Dec 21 '18 at 23:11
$begingroup$
You know what, I had a brain fart and forgot that. I think I had gotten my mind set on primitive idempotents and had some tunnel-vision.
$endgroup$
– dpb492
Jan 3 at 1:26
add a comment |
1
$begingroup$
Additively generating a set of local units is a ring-theoretic property. What exactly are you looking for?
$endgroup$
– Qiaochu Yuan
Dec 21 '18 at 23:11
$begingroup$
You know what, I had a brain fart and forgot that. I think I had gotten my mind set on primitive idempotents and had some tunnel-vision.
$endgroup$
– dpb492
Jan 3 at 1:26
1
1
$begingroup$
Additively generating a set of local units is a ring-theoretic property. What exactly are you looking for?
$endgroup$
– Qiaochu Yuan
Dec 21 '18 at 23:11
$begingroup$
Additively generating a set of local units is a ring-theoretic property. What exactly are you looking for?
$endgroup$
– Qiaochu Yuan
Dec 21 '18 at 23:11
$begingroup$
You know what, I had a brain fart and forgot that. I think I had gotten my mind set on primitive idempotents and had some tunnel-vision.
$endgroup$
– dpb492
Jan 3 at 1:26
$begingroup$
You know what, I had a brain fart and forgot that. I think I had gotten my mind set on primitive idempotents and had some tunnel-vision.
$endgroup$
– dpb492
Jan 3 at 1:26
add a comment |
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$begingroup$
Additively generating a set of local units is a ring-theoretic property. What exactly are you looking for?
$endgroup$
– Qiaochu Yuan
Dec 21 '18 at 23:11
$begingroup$
You know what, I had a brain fart and forgot that. I think I had gotten my mind set on primitive idempotents and had some tunnel-vision.
$endgroup$
– dpb492
Jan 3 at 1:26