Is there such an algebra structure?
In a ring, it holds that 1)
$$a×1=a=1×a$$ and 2) $$a×0=0=0×a$$
, so naturally, this 3) also holds
$$0×1=0=1×0$$
Is there such an algebra structure weaker than a ring so that
1) or 2)
does not hold, but 3) still holds?
abstract-algebra
asked Nov 24 at 2:04
athos
82311339
add a comment |
In a ring, it holds that 1)
$$a×1=a=1×a$$ and 2) $$a×0=0=0×a$$
, so naturally, this 3) also holds
$$0×1=0=1×0$$
Is there such an algebra structure weaker than a ring so that
1) or 2)
does not hold, but 3) still holds?
abstract-algebra
asked Nov 24 at 2:04
athos
82311339
5
Maybe they shouldn’t be called 0 and 1 without specifying their properties.
– Charlie Frohman
Nov 24 at 2:18
2
What does the symbol "$1$" refer to in such a structure if condition (1) doesn't hold?
– Clive Newstead
Nov 24 at 2:41
add a comment |
In a ring, it holds that 1)
$$a×1=a=1×a$$ and 2) $$a×0=0=0×a$$
, so naturally, this 3) also holds
$$0×1=0=1×0$$
Is there such an algebra structure weaker than a ring so that
1) or 2)
does not hold, but 3) still holds?
abstract-algebra
asked Nov 24 at 2:04
athos
82311339
In a ring, it holds that 1)
$$a×1=a=1×a$$ and 2) $$a×0=0=0×a$$
, so naturally, this 3) also holds
$$0×1=0=1×0$$
Is there such an algebra structure weaker than a ring so that
1) or 2)
does not hold, but 3) still holds?
abstract-algebra
abstract-algebra
asked Nov 24 at 2:04
athos
82311339
asked Nov 24 at 2:04
athos
82311339
edited Nov 24 at 2:30
asked Nov 24 at 2:04
athos
82311339
asked Nov 24 at 2:04
athos
82311339
asked Nov 24 at 2:04
athos
82311339
82311339
5
Maybe they shouldn’t be called 0 and 1 without specifying their properties.
– Charlie Frohman
Nov 24 at 2:18
2
What does the symbol "$1$" refer to in such a structure if condition (1) doesn't hold?
– Clive Newstead
Nov 24 at 2:41
add a comment |
5
Maybe they shouldn’t be called 0 and 1 without specifying their properties.
– Charlie Frohman
Nov 24 at 2:18
2
What does the symbol "$1$" refer to in such a structure if condition (1) doesn't hold?
– Clive Newstead
Nov 24 at 2:41
5
5
Maybe they shouldn’t be called 0 and 1 without specifying their properties.
– Charlie Frohman
Nov 24 at 2:18
Maybe they shouldn’t be called 0 and 1 without specifying their properties.
– Charlie Frohman
Nov 24 at 2:18
2
2
What does the symbol "$1$" refer to in such a structure if condition (1) doesn't hold?
– Clive Newstead
Nov 24 at 2:41
What does the symbol "$1$" refer to in such a structure if condition (1) doesn't hold?
– Clive Newstead
Nov 24 at 2:41
add a comment |
1 Answer
1
active
oldest
votes
Sure. It's called a group. Say we have a group made out of four elements, called $0,1,2,3$, where each elements is a mapping of ${0,1,2,3}to{-i,1,i,-1}$ (I did $e^{ipi (n-1)/2}$), and where $times$ is the regular complex multiplication. Then any number multiplied with "$1$" is that number, and any number multiplied with "$0$" means a multiplication by $i$. Then the second equation does not hold, but the third does, since $1$ is the identity element.
answered Nov 24 at 3:30
Andrei
11k21025
In ring 0 is the identity element of addition while 1 is the one for multiplication
– athos
Nov 24 at 3:43
And what are those supposed to be in your new structure? If they have the same properties as the ring, then you have a ring, and equation 1 is in the definition of the multiplication, and equation 2 is derived from the properties of the ring. Equation 3 is always true if 1 is true.
– Andrei
Nov 24 at 3:47
if there’s some structure that has addition and multiplication but maybe not law of distribution?
– athos
Nov 24 at 3:58
add a comment |
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1 Answer
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Sure. It's called a group. Say we have a group made out of four elements, called $0,1,2,3$, where each elements is a mapping of ${0,1,2,3}to{-i,1,i,-1}$ (I did $e^{ipi (n-1)/2}$), and where $times$ is the regular complex multiplication. Then any number multiplied with "$1$" is that number, and any number multiplied with "$0$" means a multiplication by $i$. Then the second equation does not hold, but the third does, since $1$ is the identity element.
answered Nov 24 at 3:30
Andrei
11k21025
In ring 0 is the identity element of addition while 1 is the one for multiplication
– athos
Nov 24 at 3:43
And what are those supposed to be in your new structure? If they have the same properties as the ring, then you have a ring, and equation 1 is in the definition of the multiplication, and equation 2 is derived from the properties of the ring. Equation 3 is always true if 1 is true.
– Andrei
Nov 24 at 3:47
if there’s some structure that has addition and multiplication but maybe not law of distribution?
– athos
Nov 24 at 3:58
add a comment |
Sure. It's called a group. Say we have a group made out of four elements, called $0,1,2,3$, where each elements is a mapping of ${0,1,2,3}to{-i,1,i,-1}$ (I did $e^{ipi (n-1)/2}$), and where $times$ is the regular complex multiplication. Then any number multiplied with "$1$" is that number, and any number multiplied with "$0$" means a multiplication by $i$. Then the second equation does not hold, but the third does, since $1$ is the identity element.
answered Nov 24 at 3:30
Andrei
11k21025
In ring 0 is the identity element of addition while 1 is the one for multiplication
– athos
Nov 24 at 3:43
And what are those supposed to be in your new structure? If they have the same properties as the ring, then you have a ring, and equation 1 is in the definition of the multiplication, and equation 2 is derived from the properties of the ring. Equation 3 is always true if 1 is true.
– Andrei
Nov 24 at 3:47
if there’s some structure that has addition and multiplication but maybe not law of distribution?
– athos
Nov 24 at 3:58
add a comment |
Sure. It's called a group. Say we have a group made out of four elements, called $0,1,2,3$, where each elements is a mapping of ${0,1,2,3}to{-i,1,i,-1}$ (I did $e^{ipi (n-1)/2}$), and where $times$ is the regular complex multiplication. Then any number multiplied with "$1$" is that number, and any number multiplied with "$0$" means a multiplication by $i$. Then the second equation does not hold, but the third does, since $1$ is the identity element.
answered Nov 24 at 3:30
Andrei
11k21025
Sure. It's called a group. Say we have a group made out of four elements, called $0,1,2,3$, where each elements is a mapping of ${0,1,2,3}to{-i,1,i,-1}$ (I did $e^{ipi (n-1)/2}$), and where $times$ is the regular complex multiplication. Then any number multiplied with "$1$" is that number, and any number multiplied with "$0$" means a multiplication by $i$. Then the second equation does not hold, but the third does, since $1$ is the identity element.
answered Nov 24 at 3:30
Andrei
11k21025
edited Nov 24 at 3:44
answered Nov 24 at 3:30
Andrei
11k21025
answered Nov 24 at 3:30
Andrei
11k21025
answered Nov 24 at 3:30
Andrei
11k21025
11k21025
In ring 0 is the identity element of addition while 1 is the one for multiplication
– athos
Nov 24 at 3:43
And what are those supposed to be in your new structure? If they have the same properties as the ring, then you have a ring, and equation 1 is in the definition of the multiplication, and equation 2 is derived from the properties of the ring. Equation 3 is always true if 1 is true.
– Andrei
Nov 24 at 3:47
if there’s some structure that has addition and multiplication but maybe not law of distribution?
– athos
Nov 24 at 3:58
add a comment |
In ring 0 is the identity element of addition while 1 is the one for multiplication
– athos
Nov 24 at 3:43
And what are those supposed to be in your new structure? If they have the same properties as the ring, then you have a ring, and equation 1 is in the definition of the multiplication, and equation 2 is derived from the properties of the ring. Equation 3 is always true if 1 is true.
– Andrei
Nov 24 at 3:47
if there’s some structure that has addition and multiplication but maybe not law of distribution?
– athos
Nov 24 at 3:58
In ring 0 is the identity element of addition while 1 is the one for multiplication
– athos
Nov 24 at 3:43
In ring 0 is the identity element of addition while 1 is the one for multiplication
– athos
Nov 24 at 3:43
And what are those supposed to be in your new structure? If they have the same properties as the ring, then you have a ring, and equation 1 is in the definition of the multiplication, and equation 2 is derived from the properties of the ring. Equation 3 is always true if 1 is true.
– Andrei
Nov 24 at 3:47
And what are those supposed to be in your new structure? If they have the same properties as the ring, then you have a ring, and equation 1 is in the definition of the multiplication, and equation 2 is derived from the properties of the ring. Equation 3 is always true if 1 is true.
– Andrei
Nov 24 at 3:47
if there’s some structure that has addition and multiplication but maybe not law of distribution?
– athos
Nov 24 at 3:58
if there’s some structure that has addition and multiplication but maybe not law of distribution?
– athos
Nov 24 at 3:58
add a comment |
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5
Maybe they shouldn’t be called 0 and 1 without specifying their properties.
– Charlie Frohman
Nov 24 at 2:18
2
What does the symbol "$1$" refer to in such a structure if condition (1) doesn't hold?
– Clive Newstead
Nov 24 at 2:41