Mixing
“Module” versus “Abelian Group” of the Integers, German Language usage circa 1962
0
$begingroup$
In the highly respectable BBFSK Vol I B2 we are told that the integers form a module with respect to addition. Where by module they mean a set together with an operation (+) defined in it satisfying the requirement of
- associativity and commutativity,
- a neutral element applicable to every element,
- an inverse to every element.
In a footnote they say this is also called a commutative (or Abelian) group.
That is not how I would typically enumerate the laws/axioms/properties of an Abelian group, but it is consistent with the familiar definition. The term module is otherwise unfamiliar to me. I've been told that the use of the term in this way is non-standard. Is that the case? If so, how is this similar and different from the currently accepted definition?
group-theory modules definition math-history
asked Jan 31 at 7:40
Steven HattonSteven Hatton
975422
$endgroup$
|
show 5 more comments
0
$begingroup$
In the highly respectable BBFSK Vol I B2 we are told that the integers form a module with respect to addition. Where by module they mean a set together with an operation (+) defined in it satisfying the requirement of
- associativity and commutativity,
- a neutral element applicable to every element,
- an inverse to every element.
In a footnote they say this is also called a commutative (or Abelian) group.
That is not how I would typically enumerate the laws/axioms/properties of an Abelian group, but it is consistent with the familiar definition. The term module is otherwise unfamiliar to me. I've been told that the use of the term in this way is non-standard. Is that the case? If so, how is this similar and different from the currently accepted definition?
group-theory modules definition math-history
asked Jan 31 at 7:40
Steven HattonSteven Hatton
975422
$endgroup$
$begingroup$
It's an older terminology, but it checks out.
$endgroup$
– Arthur
Jan 31 at 7:41
$begingroup$
It's a largely obsolete term, but some authors used to use "module" to denote an additive subgroup of a ring, not necessarily a subring or an ideal. It's sometimes seen in describing subgroups of a ring of algebraic integers.
$endgroup$
– Lord Shark the Unknown
Jan 31 at 7:43
$begingroup$
Of course, in German the word would be Modul.
$endgroup$
– Lord Shark the Unknown
Jan 31 at 7:46
$begingroup$
Every abelian group, and thus the integers as a group, are modules over the integers as a ring.
$endgroup$
– leibnewtz
Jan 31 at 8:13
1
$begingroup$
@StevenHatton Well, there you have it: not a big surprise an early 20th century text from a physics point of view uses a definition that's not very modern. It can, however, be worked out to be an equivalent formulation.
$endgroup$
– rschwieb
Jan 31 at 11:33
|
show 5 more comments
0
0
0
$begingroup$
In the highly respectable BBFSK Vol I B2 we are told that the integers form a module with respect to addition. Where by module they mean a set together with an operation (+) defined in it satisfying the requirement of
- associativity and commutativity,
- a neutral element applicable to every element,
- an inverse to every element.
In a footnote they say this is also called a commutative (or Abelian) group.
That is not how I would typically enumerate the laws/axioms/properties of an Abelian group, but it is consistent with the familiar definition. The term module is otherwise unfamiliar to me. I've been told that the use of the term in this way is non-standard. Is that the case? If so, how is this similar and different from the currently accepted definition?
group-theory modules definition math-history
asked Jan 31 at 7:40
Steven HattonSteven Hatton
975422
$endgroup$
In the highly respectable BBFSK Vol I B2 we are told that the integers form a module with respect to addition. Where by module they mean a set together with an operation (+) defined in it satisfying the requirement of
- associativity and commutativity,
- a neutral element applicable to every element,
- an inverse to every element.
In a footnote they say this is also called a commutative (or Abelian) group.
That is not how I would typically enumerate the laws/axioms/properties of an Abelian group, but it is consistent with the familiar definition. The term module is otherwise unfamiliar to me. I've been told that the use of the term in this way is non-standard. Is that the case? If so, how is this similar and different from the currently accepted definition?
group-theory modules definition math-history
group-theory modules definition math-history
asked Jan 31 at 7:40
Steven HattonSteven Hatton
975422
asked Jan 31 at 7:40
Steven HattonSteven Hatton
975422
edited Jan 31 at 10:12
Steven Hatton
asked Jan 31 at 7:40
Steven HattonSteven Hatton
975422
asked Jan 31 at 7:40
Steven HattonSteven Hatton
975422
asked Jan 31 at 7:40
Steven HattonSteven Hatton
975422
975422
$begingroup$
It's an older terminology, but it checks out.
$endgroup$
– Arthur
Jan 31 at 7:41
$begingroup$
It's a largely obsolete term, but some authors used to use "module" to denote an additive subgroup of a ring, not necessarily a subring or an ideal. It's sometimes seen in describing subgroups of a ring of algebraic integers.
$endgroup$
– Lord Shark the Unknown
Jan 31 at 7:43
$begingroup$
Of course, in German the word would be Modul.
$endgroup$
– Lord Shark the Unknown
Jan 31 at 7:46
$begingroup$
Every abelian group, and thus the integers as a group, are modules over the integers as a ring.
$endgroup$
– leibnewtz
Jan 31 at 8:13
1
$begingroup$
@StevenHatton Well, there you have it: not a big surprise an early 20th century text from a physics point of view uses a definition that's not very modern. It can, however, be worked out to be an equivalent formulation.
$endgroup$
– rschwieb
Jan 31 at 11:33
|
show 5 more comments
$begingroup$
It's an older terminology, but it checks out.
$endgroup$
– Arthur
Jan 31 at 7:41
$begingroup$
It's a largely obsolete term, but some authors used to use "module" to denote an additive subgroup of a ring, not necessarily a subring or an ideal. It's sometimes seen in describing subgroups of a ring of algebraic integers.
$endgroup$
– Lord Shark the Unknown
Jan 31 at 7:43
$begingroup$
Of course, in German the word would be Modul.
$endgroup$
– Lord Shark the Unknown
Jan 31 at 7:46
$begingroup$
Every abelian group, and thus the integers as a group, are modules over the integers as a ring.
$endgroup$
– leibnewtz
Jan 31 at 8:13
1
$begingroup$
@StevenHatton Well, there you have it: not a big surprise an early 20th century text from a physics point of view uses a definition that's not very modern. It can, however, be worked out to be an equivalent formulation.
$endgroup$
– rschwieb
Jan 31 at 11:33
$begingroup$
It's an older terminology, but it checks out.
$endgroup$
– Arthur
Jan 31 at 7:41
$begingroup$
It's an older terminology, but it checks out.
$endgroup$
– Arthur
Jan 31 at 7:41
$begingroup$
It's a largely obsolete term, but some authors used to use "module" to denote an additive subgroup of a ring, not necessarily a subring or an ideal. It's sometimes seen in describing subgroups of a ring of algebraic integers.
$endgroup$
– Lord Shark the Unknown
Jan 31 at 7:43
$begingroup$
It's a largely obsolete term, but some authors used to use "module" to denote an additive subgroup of a ring, not necessarily a subring or an ideal. It's sometimes seen in describing subgroups of a ring of algebraic integers.
$endgroup$
– Lord Shark the Unknown
Jan 31 at 7:43
$begingroup$
Of course, in German the word would be Modul.
$endgroup$
– Lord Shark the Unknown
Jan 31 at 7:46
$begingroup$
Of course, in German the word would be Modul.
$endgroup$
– Lord Shark the Unknown
Jan 31 at 7:46
$begingroup$
Every abelian group, and thus the integers as a group, are modules over the integers as a ring.
$endgroup$
– leibnewtz
Jan 31 at 8:13
$begingroup$
Every abelian group, and thus the integers as a group, are modules over the integers as a ring.
$endgroup$
– leibnewtz
Jan 31 at 8:13
1
1
$begingroup$
@StevenHatton Well, there you have it: not a big surprise an early 20th century text from a physics point of view uses a definition that's not very modern. It can, however, be worked out to be an equivalent formulation.
$endgroup$
– rschwieb
Jan 31 at 11:33
$begingroup$
@StevenHatton Well, there you have it: not a big surprise an early 20th century text from a physics point of view uses a definition that's not very modern. It can, however, be worked out to be an equivalent formulation.
$endgroup$
– rschwieb
Jan 31 at 11:33
|
show 5 more comments
1 Answer
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oldest
votes
3
$begingroup$
"Nowadays", and for as long as I have known, a module over a ring is a notion that generalises that of a vector space over a field.
Since apparently a vector space is an abelian group, well, there you have it.
answered Jan 31 at 8:02
Chris CusterChris Custer
14.3k3827
$endgroup$
$begingroup$
Also note that any abelian group is a module over $Bbb Z$ and vice versa.
$endgroup$
– Arthur
Jan 31 at 8:37
$begingroup$
Apparently they are showing that the set of endomorphisms of a module forms a ring with composition as the multiplicative operator. So the "over the ring" part is redundant. If I am drawing the correct conclusion, the only such set of endomorphisms of $left(mathbb{Z},+right)$ is the set of standard multiplicative products. That is, composition of the endomorphisms of the the module is identical to standard multiplication.
$endgroup$
– Steven Hatton
Jan 31 at 10:10
2
$begingroup$
@StevenHatton "over a ring" is not redundant when specifying a module structure. An abelian group can have different structures using different rings.
$endgroup$
– rschwieb
Jan 31 at 12:49
$begingroup$
@rschwieb Indeed. I see what you mean. In my current situation the module precedes the ring in terms of the order of development. And the ring is a direct manifestation of the module. Beyond that, I've reached mental saturation. I need to give this time to sink in.
$endgroup$
– Steven Hatton
Jan 31 at 13:13
add a comment |
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1 Answer
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1 Answer
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3
$begingroup$
"Nowadays", and for as long as I have known, a module over a ring is a notion that generalises that of a vector space over a field.
Since apparently a vector space is an abelian group, well, there you have it.
answered Jan 31 at 8:02
Chris CusterChris Custer
14.3k3827
$endgroup$
$begingroup$
Also note that any abelian group is a module over $Bbb Z$ and vice versa.
$endgroup$
– Arthur
Jan 31 at 8:37
$begingroup$
Apparently they are showing that the set of endomorphisms of a module forms a ring with composition as the multiplicative operator. So the "over the ring" part is redundant. If I am drawing the correct conclusion, the only such set of endomorphisms of $left(mathbb{Z},+right)$ is the set of standard multiplicative products. That is, composition of the endomorphisms of the the module is identical to standard multiplication.
$endgroup$
– Steven Hatton
Jan 31 at 10:10
2
$begingroup$
@StevenHatton "over a ring" is not redundant when specifying a module structure. An abelian group can have different structures using different rings.
$endgroup$
– rschwieb
Jan 31 at 12:49
$begingroup$
@rschwieb Indeed. I see what you mean. In my current situation the module precedes the ring in terms of the order of development. And the ring is a direct manifestation of the module. Beyond that, I've reached mental saturation. I need to give this time to sink in.
$endgroup$
– Steven Hatton
Jan 31 at 13:13
add a comment |
3
$begingroup$
"Nowadays", and for as long as I have known, a module over a ring is a notion that generalises that of a vector space over a field.
Since apparently a vector space is an abelian group, well, there you have it.
answered Jan 31 at 8:02
Chris CusterChris Custer
14.3k3827
$endgroup$
$begingroup$
Also note that any abelian group is a module over $Bbb Z$ and vice versa.
$endgroup$
– Arthur
Jan 31 at 8:37
$begingroup$
Apparently they are showing that the set of endomorphisms of a module forms a ring with composition as the multiplicative operator. So the "over the ring" part is redundant. If I am drawing the correct conclusion, the only such set of endomorphisms of $left(mathbb{Z},+right)$ is the set of standard multiplicative products. That is, composition of the endomorphisms of the the module is identical to standard multiplication.
$endgroup$
– Steven Hatton
Jan 31 at 10:10
2
$begingroup$
@StevenHatton "over a ring" is not redundant when specifying a module structure. An abelian group can have different structures using different rings.
$endgroup$
– rschwieb
Jan 31 at 12:49
$begingroup$
@rschwieb Indeed. I see what you mean. In my current situation the module precedes the ring in terms of the order of development. And the ring is a direct manifestation of the module. Beyond that, I've reached mental saturation. I need to give this time to sink in.
$endgroup$
– Steven Hatton
Jan 31 at 13:13
add a comment |
3
3
3
$begingroup$
"Nowadays", and for as long as I have known, a module over a ring is a notion that generalises that of a vector space over a field.
Since apparently a vector space is an abelian group, well, there you have it.
answered Jan 31 at 8:02
Chris CusterChris Custer
14.3k3827
$endgroup$
"Nowadays", and for as long as I have known, a module over a ring is a notion that generalises that of a vector space over a field.
Since apparently a vector space is an abelian group, well, there you have it.
answered Jan 31 at 8:02
Chris CusterChris Custer
14.3k3827
answered Jan 31 at 8:02
Chris CusterChris Custer
14.3k3827
answered Jan 31 at 8:02
Chris CusterChris Custer
14.3k3827
answered Jan 31 at 8:02
Chris CusterChris Custer
14.3k3827
14.3k3827
$begingroup$
Also note that any abelian group is a module over $Bbb Z$ and vice versa.
$endgroup$
– Arthur
Jan 31 at 8:37
$begingroup$
Apparently they are showing that the set of endomorphisms of a module forms a ring with composition as the multiplicative operator. So the "over the ring" part is redundant. If I am drawing the correct conclusion, the only such set of endomorphisms of $left(mathbb{Z},+right)$ is the set of standard multiplicative products. That is, composition of the endomorphisms of the the module is identical to standard multiplication.
$endgroup$
– Steven Hatton
Jan 31 at 10:10
2
$begingroup$
@StevenHatton "over a ring" is not redundant when specifying a module structure. An abelian group can have different structures using different rings.
$endgroup$
– rschwieb
Jan 31 at 12:49
$begingroup$
@rschwieb Indeed. I see what you mean. In my current situation the module precedes the ring in terms of the order of development. And the ring is a direct manifestation of the module. Beyond that, I've reached mental saturation. I need to give this time to sink in.
$endgroup$
– Steven Hatton
Jan 31 at 13:13
add a comment |
$begingroup$
Also note that any abelian group is a module over $Bbb Z$ and vice versa.
$endgroup$
– Arthur
Jan 31 at 8:37
$begingroup$
Apparently they are showing that the set of endomorphisms of a module forms a ring with composition as the multiplicative operator. So the "over the ring" part is redundant. If I am drawing the correct conclusion, the only such set of endomorphisms of $left(mathbb{Z},+right)$ is the set of standard multiplicative products. That is, composition of the endomorphisms of the the module is identical to standard multiplication.
$endgroup$
– Steven Hatton
Jan 31 at 10:10
2
$begingroup$
@StevenHatton "over a ring" is not redundant when specifying a module structure. An abelian group can have different structures using different rings.
$endgroup$
– rschwieb
Jan 31 at 12:49
$begingroup$
@rschwieb Indeed. I see what you mean. In my current situation the module precedes the ring in terms of the order of development. And the ring is a direct manifestation of the module. Beyond that, I've reached mental saturation. I need to give this time to sink in.
$endgroup$
– Steven Hatton
Jan 31 at 13:13
$begingroup$
Also note that any abelian group is a module over $Bbb Z$ and vice versa.
$endgroup$
– Arthur
Jan 31 at 8:37
$begingroup$
Also note that any abelian group is a module over $Bbb Z$ and vice versa.
$endgroup$
– Arthur
Jan 31 at 8:37
$begingroup$
Apparently they are showing that the set of endomorphisms of a module forms a ring with composition as the multiplicative operator. So the "over the ring" part is redundant. If I am drawing the correct conclusion, the only such set of endomorphisms of $left(mathbb{Z},+right)$ is the set of standard multiplicative products. That is, composition of the endomorphisms of the the module is identical to standard multiplication.
$endgroup$
– Steven Hatton
Jan 31 at 10:10
$begingroup$
Apparently they are showing that the set of endomorphisms of a module forms a ring with composition as the multiplicative operator. So the "over the ring" part is redundant. If I am drawing the correct conclusion, the only such set of endomorphisms of $left(mathbb{Z},+right)$ is the set of standard multiplicative products. That is, composition of the endomorphisms of the the module is identical to standard multiplication.
$endgroup$
– Steven Hatton
Jan 31 at 10:10
2
2
$begingroup$
@StevenHatton "over a ring" is not redundant when specifying a module structure. An abelian group can have different structures using different rings.
$endgroup$
– rschwieb
Jan 31 at 12:49
$begingroup$
@StevenHatton "over a ring" is not redundant when specifying a module structure. An abelian group can have different structures using different rings.
$endgroup$
– rschwieb
Jan 31 at 12:49
$begingroup$
@rschwieb Indeed. I see what you mean. In my current situation the module precedes the ring in terms of the order of development. And the ring is a direct manifestation of the module. Beyond that, I've reached mental saturation. I need to give this time to sink in.
$endgroup$
– Steven Hatton
Jan 31 at 13:13
$begingroup$
@rschwieb Indeed. I see what you mean. In my current situation the module precedes the ring in terms of the order of development. And the ring is a direct manifestation of the module. Beyond that, I've reached mental saturation. I need to give this time to sink in.
$endgroup$
– Steven Hatton
Jan 31 at 13:13
add a comment |
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$begingroup$
It's an older terminology, but it checks out.
$endgroup$
– Arthur
Jan 31 at 7:41
$begingroup$
It's a largely obsolete term, but some authors used to use "module" to denote an additive subgroup of a ring, not necessarily a subring or an ideal. It's sometimes seen in describing subgroups of a ring of algebraic integers.
$endgroup$
– Lord Shark the Unknown
Jan 31 at 7:43
$begingroup$
Of course, in German the word would be Modul.
$endgroup$
– Lord Shark the Unknown
Jan 31 at 7:46
$begingroup$
Every abelian group, and thus the integers as a group, are modules over the integers as a ring.
$endgroup$
– leibnewtz
Jan 31 at 8:13
1
$begingroup$
@StevenHatton Well, there you have it: not a big surprise an early 20th century text from a physics point of view uses a definition that's not very modern. It can, however, be worked out to be an equivalent formulation.
$endgroup$
– rschwieb
Jan 31 at 11:33