Are there any references on linear algebra (module theory) over non unital rings?
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Are there any references on linear algebra (module theory) over non unital rings? What are the main differences with unital rings in that respect?
ring-theory modules
asked Jan 15 at 22:22
Patrick SolePatrick Sole
1177
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add a comment |
$begingroup$
Are there any references on linear algebra (module theory) over non unital rings? What are the main differences with unital rings in that respect?
ring-theory modules
asked Jan 15 at 22:22
Patrick SolePatrick Sole
1177
$endgroup$
$begingroup$
The main difference is that almost nothing works. The reason is simple: defining $rx=0$ for every $rin R$ and every $xin A$ defines an $R$-module structure on the abelian group $A$. Something better happens when you require that $RA=A$, in order that $A$ is a module.
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– egreg
Jan 15 at 23:20
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Can you elaborate on this simple reason?
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– Patrick Sole
Jan 15 at 23:26
1
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@PatrickSole doesn’t it seem a little bad that every Abelian group is a (nonunitary) module over every rng? That’s certainly not the case for unitary modules.
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– rschwieb
Jan 16 at 0:47
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Things break down severely. This is more or less the reason why I insist that rings should have multiplicative neutral elements :-)
$endgroup$
– Jyrki Lahtonen
Jan 17 at 22:54
add a comment |
$begingroup$
Are there any references on linear algebra (module theory) over non unital rings? What are the main differences with unital rings in that respect?
ring-theory modules
asked Jan 15 at 22:22
Patrick SolePatrick Sole
1177
$endgroup$
Are there any references on linear algebra (module theory) over non unital rings? What are the main differences with unital rings in that respect?
ring-theory modules
ring-theory modules
asked Jan 15 at 22:22
Patrick SolePatrick Sole
1177
asked Jan 15 at 22:22
Patrick SolePatrick Sole
1177
asked Jan 15 at 22:22
Patrick SolePatrick Sole
1177
asked Jan 15 at 22:22
Patrick SolePatrick Sole
1177
asked Jan 15 at 22:22
Patrick SolePatrick Sole
1177
1177
$begingroup$
The main difference is that almost nothing works. The reason is simple: defining $rx=0$ for every $rin R$ and every $xin A$ defines an $R$-module structure on the abelian group $A$. Something better happens when you require that $RA=A$, in order that $A$ is a module.
$endgroup$
– egreg
Jan 15 at 23:20
$begingroup$
Can you elaborate on this simple reason?
$endgroup$
– Patrick Sole
Jan 15 at 23:26
1
$begingroup$
@PatrickSole doesn’t it seem a little bad that every Abelian group is a (nonunitary) module over every rng? That’s certainly not the case for unitary modules.
$endgroup$
– rschwieb
Jan 16 at 0:47
$begingroup$
Things break down severely. This is more or less the reason why I insist that rings should have multiplicative neutral elements :-)
$endgroup$
– Jyrki Lahtonen
Jan 17 at 22:54
add a comment |
$begingroup$
The main difference is that almost nothing works. The reason is simple: defining $rx=0$ for every $rin R$ and every $xin A$ defines an $R$-module structure on the abelian group $A$. Something better happens when you require that $RA=A$, in order that $A$ is a module.
$endgroup$
– egreg
Jan 15 at 23:20
$begingroup$
Can you elaborate on this simple reason?
$endgroup$
– Patrick Sole
Jan 15 at 23:26
1
$begingroup$
@PatrickSole doesn’t it seem a little bad that every Abelian group is a (nonunitary) module over every rng? That’s certainly not the case for unitary modules.
$endgroup$
– rschwieb
Jan 16 at 0:47
$begingroup$
Things break down severely. This is more or less the reason why I insist that rings should have multiplicative neutral elements :-)
$endgroup$
– Jyrki Lahtonen
Jan 17 at 22:54
$begingroup$
The main difference is that almost nothing works. The reason is simple: defining $rx=0$ for every $rin R$ and every $xin A$ defines an $R$-module structure on the abelian group $A$. Something better happens when you require that $RA=A$, in order that $A$ is a module.
$endgroup$
– egreg
Jan 15 at 23:20
$begingroup$
The main difference is that almost nothing works. The reason is simple: defining $rx=0$ for every $rin R$ and every $xin A$ defines an $R$-module structure on the abelian group $A$. Something better happens when you require that $RA=A$, in order that $A$ is a module.
$endgroup$
– egreg
Jan 15 at 23:20
$begingroup$
Can you elaborate on this simple reason?
$endgroup$
– Patrick Sole
Jan 15 at 23:26
$begingroup$
Can you elaborate on this simple reason?
$endgroup$
– Patrick Sole
Jan 15 at 23:26
1
1
$begingroup$
@PatrickSole doesn’t it seem a little bad that every Abelian group is a (nonunitary) module over every rng? That’s certainly not the case for unitary modules.
$endgroup$
– rschwieb
Jan 16 at 0:47
$begingroup$
@PatrickSole doesn’t it seem a little bad that every Abelian group is a (nonunitary) module over every rng? That’s certainly not the case for unitary modules.
$endgroup$
– rschwieb
Jan 16 at 0:47
$begingroup$
Things break down severely. This is more or less the reason why I insist that rings should have multiplicative neutral elements :-)
$endgroup$
– Jyrki Lahtonen
Jan 17 at 22:54
$begingroup$
Things break down severely. This is more or less the reason why I insist that rings should have multiplicative neutral elements :-)
$endgroup$
– Jyrki Lahtonen
Jan 17 at 22:54
add a comment |
1 Answer
1
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oldest
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Maybe not exactly what you're looking for, but these notes of Quillen say more about foundational matters regarding modules over non-unital rings than you probably thought possible!
He argues that the "correct" (I'm paraphrasing: he doesn't actually use that word) modules to consider are what he calls "firm modules".
answered Jan 16 at 10:50
Jeremy RickardJeremy Rickard
17.1k11746
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add a comment |
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1 Answer
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1 Answer
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oldest
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$begingroup$
Maybe not exactly what you're looking for, but these notes of Quillen say more about foundational matters regarding modules over non-unital rings than you probably thought possible!
He argues that the "correct" (I'm paraphrasing: he doesn't actually use that word) modules to consider are what he calls "firm modules".
answered Jan 16 at 10:50
Jeremy RickardJeremy Rickard
17.1k11746
$endgroup$
add a comment |
$begingroup$
Maybe not exactly what you're looking for, but these notes of Quillen say more about foundational matters regarding modules over non-unital rings than you probably thought possible!
He argues that the "correct" (I'm paraphrasing: he doesn't actually use that word) modules to consider are what he calls "firm modules".
answered Jan 16 at 10:50
Jeremy RickardJeremy Rickard
17.1k11746
$endgroup$
add a comment |
$begingroup$
Maybe not exactly what you're looking for, but these notes of Quillen say more about foundational matters regarding modules over non-unital rings than you probably thought possible!
He argues that the "correct" (I'm paraphrasing: he doesn't actually use that word) modules to consider are what he calls "firm modules".
answered Jan 16 at 10:50
Jeremy RickardJeremy Rickard
17.1k11746
$endgroup$
Maybe not exactly what you're looking for, but these notes of Quillen say more about foundational matters regarding modules over non-unital rings than you probably thought possible!
He argues that the "correct" (I'm paraphrasing: he doesn't actually use that word) modules to consider are what he calls "firm modules".
answered Jan 16 at 10:50
Jeremy RickardJeremy Rickard
17.1k11746
answered Jan 16 at 10:50
Jeremy RickardJeremy Rickard
17.1k11746
answered Jan 16 at 10:50
Jeremy RickardJeremy Rickard
17.1k11746
answered Jan 16 at 10:50
Jeremy RickardJeremy Rickard
17.1k11746
17.1k11746
add a comment |
add a comment |
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$begingroup$
The main difference is that almost nothing works. The reason is simple: defining $rx=0$ for every $rin R$ and every $xin A$ defines an $R$-module structure on the abelian group $A$. Something better happens when you require that $RA=A$, in order that $A$ is a module.
$endgroup$
– egreg
Jan 15 at 23:20
$begingroup$
Can you elaborate on this simple reason?
$endgroup$
– Patrick Sole
Jan 15 at 23:26
1
$begingroup$
@PatrickSole doesn’t it seem a little bad that every Abelian group is a (nonunitary) module over every rng? That’s certainly not the case for unitary modules.
$endgroup$
– rschwieb
Jan 16 at 0:47
$begingroup$
Things break down severely. This is more or less the reason why I insist that rings should have multiplicative neutral elements :-)
$endgroup$
– Jyrki Lahtonen
Jan 17 at 22:54