Why do we define modules only on additive groups?
$begingroup$
By definition, modules are additive abelian groups that satisfy a few properties. I know why abelian property is necessary as it is forced by its axioms. But why can we do the same with multiplicative groups, like $R^*$ or $S^1$ (mult group of roots of unity) or circle group etc.
soft-question modules
asked Jan 23 at 12:20
blablablabla
449211
$endgroup$
add a comment |
$begingroup$
By definition, modules are additive abelian groups that satisfy a few properties. I know why abelian property is necessary as it is forced by its axioms. But why can we do the same with multiplicative groups, like $R^*$ or $S^1$ (mult group of roots of unity) or circle group etc.
soft-question modules
asked Jan 23 at 12:20
blablablabla
449211
$endgroup$
1
$begingroup$
additive group is frequently just another term for abelian group. The group operation does not really had to be "addition".
$endgroup$
– lhf
Jan 23 at 12:26
$begingroup$
Multiplicative groups of number fields are occasionally viewed as modules over the group ring of the Galois group.
$endgroup$
– Jyrki Lahtonen
Jan 24 at 22:29
add a comment |
$begingroup$
By definition, modules are additive abelian groups that satisfy a few properties. I know why abelian property is necessary as it is forced by its axioms. But why can we do the same with multiplicative groups, like $R^*$ or $S^1$ (mult group of roots of unity) or circle group etc.
soft-question modules
asked Jan 23 at 12:20
blablablabla
449211
$endgroup$
By definition, modules are additive abelian groups that satisfy a few properties. I know why abelian property is necessary as it is forced by its axioms. But why can we do the same with multiplicative groups, like $R^*$ or $S^1$ (mult group of roots of unity) or circle group etc.
soft-question modules
soft-question modules
asked Jan 23 at 12:20
blablablabla
449211
asked Jan 23 at 12:20
blablablabla
449211
asked Jan 23 at 12:20
blablablabla
449211
asked Jan 23 at 12:20
blablablabla
449211
asked Jan 23 at 12:20
blablablabla
449211
449211
1
$begingroup$
additive group is frequently just another term for abelian group. The group operation does not really had to be "addition".
$endgroup$
– lhf
Jan 23 at 12:26
$begingroup$
Multiplicative groups of number fields are occasionally viewed as modules over the group ring of the Galois group.
$endgroup$
– Jyrki Lahtonen
Jan 24 at 22:29
add a comment |
1
$begingroup$
additive group is frequently just another term for abelian group. The group operation does not really had to be "addition".
$endgroup$
– lhf
Jan 23 at 12:26
$begingroup$
Multiplicative groups of number fields are occasionally viewed as modules over the group ring of the Galois group.
$endgroup$
– Jyrki Lahtonen
Jan 24 at 22:29
1
1
$begingroup$
additive group is frequently just another term for abelian group. The group operation does not really had to be "addition".
$endgroup$
– lhf
Jan 23 at 12:26
$begingroup$
additive group is frequently just another term for abelian group. The group operation does not really had to be "addition".
$endgroup$
– lhf
Jan 23 at 12:26
$begingroup$
Multiplicative groups of number fields are occasionally viewed as modules over the group ring of the Galois group.
$endgroup$
– Jyrki Lahtonen
Jan 24 at 22:29
$begingroup$
Multiplicative groups of number fields are occasionally viewed as modules over the group ring of the Galois group.
$endgroup$
– Jyrki Lahtonen
Jan 24 at 22:29
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
The notational reason is that we introduce a new operation for modules: multiplication by integers. In order to avoid confusion with the old operation, belonging to the abelian group, we denote the old operation by addition.
These two operations do come with a distributive property, so they behave like addition and multiplication "should" together.
And - well, in the big picture, if we have an abelian group, it doesn't matter what we call that group's operation. We could call the circle group's operation addition if we felt like it - it is addition for angles, after all. What we call the operation only matters if we're embedding the group in some structure with more operations, like putting the circle group into the field $mathbb{C}$ as the elements with absolute value $1$.
answered Jan 23 at 12:33
jmerryjmerry
11.8k1528
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add a comment |
$begingroup$
Well, you could take the set of positive real numbers and define $x+y = x*y$ (addition) and $cx=x^c$ (scalar multiplication). This gives you a vector space. (You basically consider the logarithm).
answered Jan 23 at 12:23
WuestenfuxWuestenfux
4,7941513
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2 Answers
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active
oldest
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2 Answers
2
active
oldest
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votes
$begingroup$
The notational reason is that we introduce a new operation for modules: multiplication by integers. In order to avoid confusion with the old operation, belonging to the abelian group, we denote the old operation by addition.
These two operations do come with a distributive property, so they behave like addition and multiplication "should" together.
And - well, in the big picture, if we have an abelian group, it doesn't matter what we call that group's operation. We could call the circle group's operation addition if we felt like it - it is addition for angles, after all. What we call the operation only matters if we're embedding the group in some structure with more operations, like putting the circle group into the field $mathbb{C}$ as the elements with absolute value $1$.
answered Jan 23 at 12:33
jmerryjmerry
11.8k1528
$endgroup$
add a comment |
$begingroup$
The notational reason is that we introduce a new operation for modules: multiplication by integers. In order to avoid confusion with the old operation, belonging to the abelian group, we denote the old operation by addition.
These two operations do come with a distributive property, so they behave like addition and multiplication "should" together.
And - well, in the big picture, if we have an abelian group, it doesn't matter what we call that group's operation. We could call the circle group's operation addition if we felt like it - it is addition for angles, after all. What we call the operation only matters if we're embedding the group in some structure with more operations, like putting the circle group into the field $mathbb{C}$ as the elements with absolute value $1$.
answered Jan 23 at 12:33
jmerryjmerry
11.8k1528
$endgroup$
add a comment |
$begingroup$
The notational reason is that we introduce a new operation for modules: multiplication by integers. In order to avoid confusion with the old operation, belonging to the abelian group, we denote the old operation by addition.
These two operations do come with a distributive property, so they behave like addition and multiplication "should" together.
And - well, in the big picture, if we have an abelian group, it doesn't matter what we call that group's operation. We could call the circle group's operation addition if we felt like it - it is addition for angles, after all. What we call the operation only matters if we're embedding the group in some structure with more operations, like putting the circle group into the field $mathbb{C}$ as the elements with absolute value $1$.
answered Jan 23 at 12:33
jmerryjmerry
11.8k1528
$endgroup$
The notational reason is that we introduce a new operation for modules: multiplication by integers. In order to avoid confusion with the old operation, belonging to the abelian group, we denote the old operation by addition.
These two operations do come with a distributive property, so they behave like addition and multiplication "should" together.
And - well, in the big picture, if we have an abelian group, it doesn't matter what we call that group's operation. We could call the circle group's operation addition if we felt like it - it is addition for angles, after all. What we call the operation only matters if we're embedding the group in some structure with more operations, like putting the circle group into the field $mathbb{C}$ as the elements with absolute value $1$.
answered Jan 23 at 12:33
jmerryjmerry
11.8k1528
answered Jan 23 at 12:33
jmerryjmerry
11.8k1528
answered Jan 23 at 12:33
jmerryjmerry
11.8k1528
answered Jan 23 at 12:33
jmerryjmerry
11.8k1528
11.8k1528
add a comment |
add a comment |
$begingroup$
Well, you could take the set of positive real numbers and define $x+y = x*y$ (addition) and $cx=x^c$ (scalar multiplication). This gives you a vector space. (You basically consider the logarithm).
answered Jan 23 at 12:23
WuestenfuxWuestenfux
4,7941513
$endgroup$
add a comment |
$begingroup$
Well, you could take the set of positive real numbers and define $x+y = x*y$ (addition) and $cx=x^c$ (scalar multiplication). This gives you a vector space. (You basically consider the logarithm).
answered Jan 23 at 12:23
WuestenfuxWuestenfux
4,7941513
$endgroup$
add a comment |
$begingroup$
Well, you could take the set of positive real numbers and define $x+y = x*y$ (addition) and $cx=x^c$ (scalar multiplication). This gives you a vector space. (You basically consider the logarithm).
answered Jan 23 at 12:23
WuestenfuxWuestenfux
4,7941513
$endgroup$
Well, you could take the set of positive real numbers and define $x+y = x*y$ (addition) and $cx=x^c$ (scalar multiplication). This gives you a vector space. (You basically consider the logarithm).
answered Jan 23 at 12:23
WuestenfuxWuestenfux
4,7941513
answered Jan 23 at 12:23
WuestenfuxWuestenfux
4,7941513
answered Jan 23 at 12:23
WuestenfuxWuestenfux
4,7941513
answered Jan 23 at 12:23
WuestenfuxWuestenfux
4,7941513
4,7941513
add a comment |
add a comment |
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1
$begingroup$
additive group is frequently just another term for abelian group. The group operation does not really had to be "addition".
$endgroup$
– lhf
Jan 23 at 12:26
$begingroup$
Multiplicative groups of number fields are occasionally viewed as modules over the group ring of the Galois group.
$endgroup$
– Jyrki Lahtonen
Jan 24 at 22:29