About the definition of subring
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Reading Atiyah-MacDonald: Introduction to Commutative Algebra, I found the following definition of subring:
A subset $S$ of a ring $A$ is a subring of $A$ if $S$ is closed under addition and multiplication and contains the identity element of $A$. The identity mapping of S into A is then a ring homomorphism.
I know this definition is "wrong", as on the question I linked below is said:
Concept of a subring in Atiyah-Macdonald's book
But my question is, what if we change our "classic" definition of subring by this other? For me, it seems that everything remains equal and, at least, in the context of rings, there is not any contradiction.
For example, the following property is still true:
If $f: A rightarrow B$ is a ring homomorphism, then $operatorname{im}(f) subset B$ is a subring.
I can't find any problem with this redefinition of subring. It will be welcome any correction or comment. Thanks everyone!
ring-theory commutative-algebra definition
edited Nov 15 at 11:15
Bernard
116k637108
asked Nov 15 at 11:12
AlgebraicallyClosed
10311
add a comment |
up vote
0
down vote
favorite
Reading Atiyah-MacDonald: Introduction to Commutative Algebra, I found the following definition of subring:
A subset $S$ of a ring $A$ is a subring of $A$ if $S$ is closed under addition and multiplication and contains the identity element of $A$. The identity mapping of S into A is then a ring homomorphism.
I know this definition is "wrong", as on the question I linked below is said:
Concept of a subring in Atiyah-Macdonald's book
But my question is, what if we change our "classic" definition of subring by this other? For me, it seems that everything remains equal and, at least, in the context of rings, there is not any contradiction.
For example, the following property is still true:
If $f: A rightarrow B$ is a ring homomorphism, then $operatorname{im}(f) subset B$ is a subring.
I can't find any problem with this redefinition of subring. It will be welcome any correction or comment. Thanks everyone!
ring-theory commutative-algebra definition
edited Nov 15 at 11:15
Bernard
116k637108
asked Nov 15 at 11:12
AlgebraicallyClosed
10311
Sure, changing definitions won't give you a contradiction. It's just that a "subring" of a ring now isn't necessarily a ring, which is ridiculous (for example, $mathbb{N}$ is a "subring" of $mathbb{Z}$).
– user3482749
Nov 15 at 11:21
I understand what you say @user3482749. But if we don't name these structures "subrings" and see them as another possible substructure of rings, what are we obtaining? For example, we don't require ideals to be again rings.
– AlgebraicallyClosed
Nov 15 at 11:30
A sub-semiring.
– user3482749
Nov 15 at 11:33
Is this fact suggesting that our theorems could be weakened because all of them are still valid? @user3482749
– AlgebraicallyClosed
Nov 15 at 11:50
1
No. A great many statements about rings are not true about semirings.
– user3482749
Nov 15 at 11:59
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Reading Atiyah-MacDonald: Introduction to Commutative Algebra, I found the following definition of subring:
A subset $S$ of a ring $A$ is a subring of $A$ if $S$ is closed under addition and multiplication and contains the identity element of $A$. The identity mapping of S into A is then a ring homomorphism.
I know this definition is "wrong", as on the question I linked below is said:
Concept of a subring in Atiyah-Macdonald's book
But my question is, what if we change our "classic" definition of subring by this other? For me, it seems that everything remains equal and, at least, in the context of rings, there is not any contradiction.
For example, the following property is still true:
If $f: A rightarrow B$ is a ring homomorphism, then $operatorname{im}(f) subset B$ is a subring.
I can't find any problem with this redefinition of subring. It will be welcome any correction or comment. Thanks everyone!
ring-theory commutative-algebra definition
edited Nov 15 at 11:15
Bernard
116k637108
asked Nov 15 at 11:12
AlgebraicallyClosed
10311
Reading Atiyah-MacDonald: Introduction to Commutative Algebra, I found the following definition of subring:
A subset $S$ of a ring $A$ is a subring of $A$ if $S$ is closed under addition and multiplication and contains the identity element of $A$. The identity mapping of S into A is then a ring homomorphism.
I know this definition is "wrong", as on the question I linked below is said:
Concept of a subring in Atiyah-Macdonald's book
But my question is, what if we change our "classic" definition of subring by this other? For me, it seems that everything remains equal and, at least, in the context of rings, there is not any contradiction.
For example, the following property is still true:
If $f: A rightarrow B$ is a ring homomorphism, then $operatorname{im}(f) subset B$ is a subring.
I can't find any problem with this redefinition of subring. It will be welcome any correction or comment. Thanks everyone!
ring-theory commutative-algebra definition
ring-theory commutative-algebra definition
edited Nov 15 at 11:15
Bernard
116k637108
asked Nov 15 at 11:12
AlgebraicallyClosed
10311
edited Nov 15 at 11:15
Bernard
116k637108
asked Nov 15 at 11:12
AlgebraicallyClosed
10311
edited Nov 15 at 11:15
Bernard
116k637108
edited Nov 15 at 11:15
Bernard
116k637108
edited Nov 15 at 11:15
Bernard
116k637108
116k637108
asked Nov 15 at 11:12
AlgebraicallyClosed
10311
asked Nov 15 at 11:12
AlgebraicallyClosed
10311
asked Nov 15 at 11:12
AlgebraicallyClosed
10311
10311
Sure, changing definitions won't give you a contradiction. It's just that a "subring" of a ring now isn't necessarily a ring, which is ridiculous (for example, $mathbb{N}$ is a "subring" of $mathbb{Z}$).
– user3482749
Nov 15 at 11:21
I understand what you say @user3482749. But if we don't name these structures "subrings" and see them as another possible substructure of rings, what are we obtaining? For example, we don't require ideals to be again rings.
– AlgebraicallyClosed
Nov 15 at 11:30
A sub-semiring.
– user3482749
Nov 15 at 11:33
Is this fact suggesting that our theorems could be weakened because all of them are still valid? @user3482749
– AlgebraicallyClosed
Nov 15 at 11:50
1
No. A great many statements about rings are not true about semirings.
– user3482749
Nov 15 at 11:59
add a comment |
Sure, changing definitions won't give you a contradiction. It's just that a "subring" of a ring now isn't necessarily a ring, which is ridiculous (for example, $mathbb{N}$ is a "subring" of $mathbb{Z}$).
– user3482749
Nov 15 at 11:21
I understand what you say @user3482749. But if we don't name these structures "subrings" and see them as another possible substructure of rings, what are we obtaining? For example, we don't require ideals to be again rings.
– AlgebraicallyClosed
Nov 15 at 11:30
A sub-semiring.
– user3482749
Nov 15 at 11:33
Is this fact suggesting that our theorems could be weakened because all of them are still valid? @user3482749
– AlgebraicallyClosed
Nov 15 at 11:50
1
No. A great many statements about rings are not true about semirings.
– user3482749
Nov 15 at 11:59
Sure, changing definitions won't give you a contradiction. It's just that a "subring" of a ring now isn't necessarily a ring, which is ridiculous (for example, $mathbb{N}$ is a "subring" of $mathbb{Z}$).
– user3482749
Nov 15 at 11:21
Sure, changing definitions won't give you a contradiction. It's just that a "subring" of a ring now isn't necessarily a ring, which is ridiculous (for example, $mathbb{N}$ is a "subring" of $mathbb{Z}$).
– user3482749
Nov 15 at 11:21
I understand what you say @user3482749. But if we don't name these structures "subrings" and see them as another possible substructure of rings, what are we obtaining? For example, we don't require ideals to be again rings.
– AlgebraicallyClosed
Nov 15 at 11:30
I understand what you say @user3482749. But if we don't name these structures "subrings" and see them as another possible substructure of rings, what are we obtaining? For example, we don't require ideals to be again rings.
– AlgebraicallyClosed
Nov 15 at 11:30
A sub-semiring.
– user3482749
Nov 15 at 11:33
A sub-semiring.
– user3482749
Nov 15 at 11:33
Is this fact suggesting that our theorems could be weakened because all of them are still valid? @user3482749
– AlgebraicallyClosed
Nov 15 at 11:50
Is this fact suggesting that our theorems could be weakened because all of them are still valid? @user3482749
– AlgebraicallyClosed
Nov 15 at 11:50
1
1
No. A great many statements about rings are not true about semirings.
– user3482749
Nov 15 at 11:59
No. A great many statements about rings are not true about semirings.
– user3482749
Nov 15 at 11:59
add a comment |
1 Answer
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oldest
votes
up vote
0
down vote
For clarity, let's use the following definitions (#-subring is not a standard term, just using it to distinguish the two possible definitions of subring).
A subset $S$ of a ring $A$ is a #-subring of $A$ if $S$ is closed under addition and multiplication and contains the identity element of $A$.
A subset $S$ of a ring $A$ is a subring of $A$ if $S$ is an additive subgroup of $A$, is closed under multiplication and contains the identity element of $A$.
Then any subring is a #-subring, so the property you've given clearly still holds. As will anything that proves some subset is a subring.
The problem is that a #-subring is not necessarily a ring: $Bbb N$ is a #-subring of $Bbb Z$, yet $Bbb N$ is not a ring. Hopefully it is clear to you why a subring has to be a ring for it to be a sensible choice of definition!
A #-subring is, however, a semiring (a semiring has the same definition as a ring, but without the requirement for additive inverses - so in particular, any ring is a semiring), and in fact a #-subring of a ring $R$ is precisely a subsemiring of the ring (and thus semiring) $R$.
answered Nov 15 at 11:28
Christopher
6,27811628
As I said in the response of the comment above, I understand the incoherence of call something a subring that is not itself a ring, but my question is, What structure has defined Atiyah and why all the theorems could be enunciated in these "new" terms?
– AlgebraicallyClosed
Nov 15 at 11:38
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
For clarity, let's use the following definitions (#-subring is not a standard term, just using it to distinguish the two possible definitions of subring).
A subset $S$ of a ring $A$ is a #-subring of $A$ if $S$ is closed under addition and multiplication and contains the identity element of $A$.
A subset $S$ of a ring $A$ is a subring of $A$ if $S$ is an additive subgroup of $A$, is closed under multiplication and contains the identity element of $A$.
Then any subring is a #-subring, so the property you've given clearly still holds. As will anything that proves some subset is a subring.
The problem is that a #-subring is not necessarily a ring: $Bbb N$ is a #-subring of $Bbb Z$, yet $Bbb N$ is not a ring. Hopefully it is clear to you why a subring has to be a ring for it to be a sensible choice of definition!
A #-subring is, however, a semiring (a semiring has the same definition as a ring, but without the requirement for additive inverses - so in particular, any ring is a semiring), and in fact a #-subring of a ring $R$ is precisely a subsemiring of the ring (and thus semiring) $R$.
answered Nov 15 at 11:28
Christopher
6,27811628
As I said in the response of the comment above, I understand the incoherence of call something a subring that is not itself a ring, but my question is, What structure has defined Atiyah and why all the theorems could be enunciated in these "new" terms?
– AlgebraicallyClosed
Nov 15 at 11:38
add a comment |
up vote
0
down vote
For clarity, let's use the following definitions (#-subring is not a standard term, just using it to distinguish the two possible definitions of subring).
A subset $S$ of a ring $A$ is a #-subring of $A$ if $S$ is closed under addition and multiplication and contains the identity element of $A$.
A subset $S$ of a ring $A$ is a subring of $A$ if $S$ is an additive subgroup of $A$, is closed under multiplication and contains the identity element of $A$.
Then any subring is a #-subring, so the property you've given clearly still holds. As will anything that proves some subset is a subring.
The problem is that a #-subring is not necessarily a ring: $Bbb N$ is a #-subring of $Bbb Z$, yet $Bbb N$ is not a ring. Hopefully it is clear to you why a subring has to be a ring for it to be a sensible choice of definition!
A #-subring is, however, a semiring (a semiring has the same definition as a ring, but without the requirement for additive inverses - so in particular, any ring is a semiring), and in fact a #-subring of a ring $R$ is precisely a subsemiring of the ring (and thus semiring) $R$.
answered Nov 15 at 11:28
Christopher
6,27811628
As I said in the response of the comment above, I understand the incoherence of call something a subring that is not itself a ring, but my question is, What structure has defined Atiyah and why all the theorems could be enunciated in these "new" terms?
– AlgebraicallyClosed
Nov 15 at 11:38
add a comment |
up vote
0
down vote
up vote
0
down vote
For clarity, let's use the following definitions (#-subring is not a standard term, just using it to distinguish the two possible definitions of subring).
A subset $S$ of a ring $A$ is a #-subring of $A$ if $S$ is closed under addition and multiplication and contains the identity element of $A$.
A subset $S$ of a ring $A$ is a subring of $A$ if $S$ is an additive subgroup of $A$, is closed under multiplication and contains the identity element of $A$.
Then any subring is a #-subring, so the property you've given clearly still holds. As will anything that proves some subset is a subring.
The problem is that a #-subring is not necessarily a ring: $Bbb N$ is a #-subring of $Bbb Z$, yet $Bbb N$ is not a ring. Hopefully it is clear to you why a subring has to be a ring for it to be a sensible choice of definition!
A #-subring is, however, a semiring (a semiring has the same definition as a ring, but without the requirement for additive inverses - so in particular, any ring is a semiring), and in fact a #-subring of a ring $R$ is precisely a subsemiring of the ring (and thus semiring) $R$.
answered Nov 15 at 11:28
Christopher
6,27811628
For clarity, let's use the following definitions (#-subring is not a standard term, just using it to distinguish the two possible definitions of subring).
A subset $S$ of a ring $A$ is a #-subring of $A$ if $S$ is closed under addition and multiplication and contains the identity element of $A$.
A subset $S$ of a ring $A$ is a subring of $A$ if $S$ is an additive subgroup of $A$, is closed under multiplication and contains the identity element of $A$.
Then any subring is a #-subring, so the property you've given clearly still holds. As will anything that proves some subset is a subring.
The problem is that a #-subring is not necessarily a ring: $Bbb N$ is a #-subring of $Bbb Z$, yet $Bbb N$ is not a ring. Hopefully it is clear to you why a subring has to be a ring for it to be a sensible choice of definition!
A #-subring is, however, a semiring (a semiring has the same definition as a ring, but without the requirement for additive inverses - so in particular, any ring is a semiring), and in fact a #-subring of a ring $R$ is precisely a subsemiring of the ring (and thus semiring) $R$.
answered Nov 15 at 11:28
Christopher
6,27811628
edited Nov 15 at 11:47
answered Nov 15 at 11:28
Christopher
6,27811628
answered Nov 15 at 11:28
Christopher
6,27811628
answered Nov 15 at 11:28
Christopher
6,27811628
6,27811628
As I said in the response of the comment above, I understand the incoherence of call something a subring that is not itself a ring, but my question is, What structure has defined Atiyah and why all the theorems could be enunciated in these "new" terms?
– AlgebraicallyClosed
Nov 15 at 11:38
add a comment |
As I said in the response of the comment above, I understand the incoherence of call something a subring that is not itself a ring, but my question is, What structure has defined Atiyah and why all the theorems could be enunciated in these "new" terms?
– AlgebraicallyClosed
Nov 15 at 11:38
As I said in the response of the comment above, I understand the incoherence of call something a subring that is not itself a ring, but my question is, What structure has defined Atiyah and why all the theorems could be enunciated in these "new" terms?
– AlgebraicallyClosed
Nov 15 at 11:38
As I said in the response of the comment above, I understand the incoherence of call something a subring that is not itself a ring, but my question is, What structure has defined Atiyah and why all the theorems could be enunciated in these "new" terms?
– AlgebraicallyClosed
Nov 15 at 11:38
add a comment |
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Sure, changing definitions won't give you a contradiction. It's just that a "subring" of a ring now isn't necessarily a ring, which is ridiculous (for example, $mathbb{N}$ is a "subring" of $mathbb{Z}$).
– user3482749
Nov 15 at 11:21
I understand what you say @user3482749. But if we don't name these structures "subrings" and see them as another possible substructure of rings, what are we obtaining? For example, we don't require ideals to be again rings.
– AlgebraicallyClosed
Nov 15 at 11:30
A sub-semiring.
– user3482749
Nov 15 at 11:33
Is this fact suggesting that our theorems could be weakened because all of them are still valid? @user3482749
– AlgebraicallyClosed
Nov 15 at 11:50
1
No. A great many statements about rings are not true about semirings.
– user3482749
Nov 15 at 11:59