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Looking at the interesting list of ring properties that are inherited from a ring $mathcal{R}$ by its polynomial ring $mathcal{R}$[X] and remembering a question I once asked I want to repeat the latter in a more general way:
Can you give ring properties with catchy categorical
characterizations like these:
A ring $mathcal{R}$ has the structure of $mathbb{Z}$ iff it is an initial object in the category of rings.
A ring $mathcal{R}$ has characteristic $0$ iff the morphism from $mathbb{Z}$ is a monomorphism.
What about being commutative, factorial, Noetherian, Abelian, or an integral domain?
[Note that the property of having a multiplicative identity (i.e. of being unital) doesn’t have to be defined, because it’s presupposed in the category of rings.]
List of characterizations from the answers below:

A ring $mathcal{R}$ is Noetherian iff every ascending chain of strong epimorphisms is stationary.

A ring $mathcal{R}$ is finitely presented iff it is a compact object.
add a commentÂ Â
6
down vote
favorite
Looking at the interesting list of ring properties that are inherited from a ring $mathcal{R}$ by its polynomial ring $mathcal{R}$[X] and remembering a question I once asked I want to repeat the latter in a more general way:
Can you give ring properties with catchy categorical
characterizations like these:
A ring $mathcal{R}$ has the structure of $mathbb{Z}$ iff it is an initial object in the category of rings.
A ring $mathcal{R}$ has characteristic $0$ iff the morphism from $mathbb{Z}$ is a monomorphism.
What about being commutative, factorial, Noetherian, Abelian, or an integral domain?
[Note that the property of having a multiplicative identity (i.e. of being unital) doesn’t have to be defined, because it’s presupposed in the category of rings.]
List of characterizations from the answers below:

A ring $mathcal{R}$ is Noetherian iff every ascending chain of strong epimorphisms is stationary.

A ring $mathcal{R}$ is finitely presented iff it is a compact object.
add a commentÂ Â
6
down vote
favorite
6
down vote
favorite
Looking at the interesting list of ring properties that are inherited from a ring $mathcal{R}$ by its polynomial ring $mathcal{R}$[X] and remembering a question I once asked I want to repeat the latter in a more general way:
Can you give ring properties with catchy categorical
characterizations like these:
A ring $mathcal{R}$ has the structure of $mathbb{Z}$ iff it is an initial object in the category of rings.
A ring $mathcal{R}$ has characteristic $0$ iff the morphism from $mathbb{Z}$ is a monomorphism.
What about being commutative, factorial, Noetherian, Abelian, or an integral domain?
[Note that the property of having a multiplicative identity (i.e. of being unital) doesn’t have to be defined, because it’s presupposed in the category of rings.]
List of characterizations from the answers below:

A ring $mathcal{R}$ is Noetherian iff every ascending chain of strong epimorphisms is stationary.

A ring $mathcal{R}$ is finitely presented iff it is a compact object.
Looking at the interesting list of ring properties that are inherited from a ring $mathcal{R}$ by its polynomial ring $mathcal{R}$[X] and remembering a question I once asked I want to repeat the latter in a more general way:
Can you give ring properties with catchy categorical
characterizations like these:
A ring $mathcal{R}$ has the structure of $mathbb{Z}$ iff it is an initial object in the category of rings.
A ring $mathcal{R}$ has characteristic $0$ iff the morphism from $mathbb{Z}$ is a monomorphism.
What about being commutative, factorial, Noetherian, Abelian, or an integral domain?
[Note that the property of having a multiplicative identity (i.e. of being unital) doesn’t have to be defined, because it’s presupposed in the category of rings.]
List of characterizations from the answers below:

A ring $mathcal{R}$ is Noetherian iff every ascending chain of strong epimorphisms is stationary.

A ring $mathcal{R}$ is finitely presented iff it is a compact object.
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2 Answers
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There is a notion of noetherian object in any category, which is simply that every ascending chain of subobjects is stationnary. With this definition, noetherian objects in the category of modules over a ring are simply noetherian modules; however, ideals are not subrings (since in general they do not contain the identity element), so noetherian objects in the category of unital rings are not noetherian rings.
This is easy to fix, however : every ascending chain of ideals
$$I_0subset I_1subsetdots subset I_{n}subset I_{n+1}subset dots,$$
in a ring $R$ induces an ascending chain of quotient rings
$$R/I_0to R/I_1todotsto R/I_nto R/I_{n+1}to dots$$
and the original chain of ideals is stationnary if and only if the chain of quotients is. Now quotient maps are just the surjective maps, which are the same thing as strong or regular epimorphisms in the category of rings (or in any “algebraic” category). Thus noetherian rings are precisely those for which every ascending chain of strong epimorphisms is stationnary, which one might call “strongly conoetherian objects”.
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1
down vote
The finitely presented rings are the compact objects of the category of rings.
That is, those objects $R$ such that the functor $hom(R,)colon mathrm{(Ring)}tomathrm{(Set)}$ preserves filtered colimits.
In fact, this applies to any variety of algebras, e.g., by JiÃ
ÂÃÂ AdÃÂ¡mek, JiÃ
ÂÃÂ RosickÃÂ½, Locally Presentable and Accessible Categories, Corollary 3.13.

Also see qchu.wordpress.com/2015/04/25/compactobjects and replace modules by rings for a proof.
âÂ Ben
22 mins ago
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
4
down vote
There is a notion of noetherian object in any category, which is simply that every ascending chain of subobjects is stationnary. With this definition, noetherian objects in the category of modules over a ring are simply noetherian modules; however, ideals are not subrings (since in general they do not contain the identity element), so noetherian objects in the category of unital rings are not noetherian rings.
This is easy to fix, however : every ascending chain of ideals
$$I_0subset I_1subsetdots subset I_{n}subset I_{n+1}subset dots,$$
in a ring $R$ induces an ascending chain of quotient rings
$$R/I_0to R/I_1todotsto R/I_nto R/I_{n+1}to dots$$
and the original chain of ideals is stationnary if and only if the chain of quotients is. Now quotient maps are just the surjective maps, which are the same thing as strong or regular epimorphisms in the category of rings (or in any “algebraic” category). Thus noetherian rings are precisely those for which every ascending chain of strong epimorphisms is stationnary, which one might call “strongly conoetherian objects”.
add a commentÂ Â
4
down vote
There is a notion of noetherian object in any category, which is simply that every ascending chain of subobjects is stationnary. With this definition, noetherian objects in the category of modules over a ring are simply noetherian modules; however, ideals are not subrings (since in general they do not contain the identity element), so noetherian objects in the category of unital rings are not noetherian rings.
This is easy to fix, however : every ascending chain of ideals
$$I_0subset I_1subsetdots subset I_{n}subset I_{n+1}subset dots,$$
in a ring $R$ induces an ascending chain of quotient rings
$$R/I_0to R/I_1todotsto R/I_nto R/I_{n+1}to dots$$
and the original chain of ideals is stationnary if and only if the chain of quotients is. Now quotient maps are just the surjective maps, which are the same thing as strong or regular epimorphisms in the category of rings (or in any “algebraic” category). Thus noetherian rings are precisely those for which every ascending chain of strong epimorphisms is stationnary, which one might call “strongly conoetherian objects”.
add a commentÂ Â
4
down vote
4
down vote
There is a notion of noetherian object in any category, which is simply that every ascending chain of subobjects is stationnary. With this definition, noetherian objects in the category of modules over a ring are simply noetherian modules; however, ideals are not subrings (since in general they do not contain the identity element), so noetherian objects in the category of unital rings are not noetherian rings.
This is easy to fix, however : every ascending chain of ideals
$$I_0subset I_1subsetdots subset I_{n}subset I_{n+1}subset dots,$$
in a ring $R$ induces an ascending chain of quotient rings
$$R/I_0to R/I_1todotsto R/I_nto R/I_{n+1}to dots$$
and the original chain of ideals is stationnary if and only if the chain of quotients is. Now quotient maps are just the surjective maps, which are the same thing as strong or regular epimorphisms in the category of rings (or in any “algebraic” category). Thus noetherian rings are precisely those for which every ascending chain of strong epimorphisms is stationnary, which one might call “strongly conoetherian objects”.
There is a notion of noetherian object in any category, which is simply that every ascending chain of subobjects is stationnary. With this definition, noetherian objects in the category of modules over a ring are simply noetherian modules; however, ideals are not subrings (since in general they do not contain the identity element), so noetherian objects in the category of unital rings are not noetherian rings.
This is easy to fix, however : every ascending chain of ideals
$$I_0subset I_1subsetdots subset I_{n}subset I_{n+1}subset dots,$$
in a ring $R$ induces an ascending chain of quotient rings
$$R/I_0to R/I_1todotsto R/I_nto R/I_{n+1}to dots$$
and the original chain of ideals is stationnary if and only if the chain of quotients is. Now quotient maps are just the surjective maps, which are the same thing as strong or regular epimorphisms in the category of rings (or in any “algebraic” category). Thus noetherian rings are precisely those for which every ascending chain of strong epimorphisms is stationnary, which one might call “strongly conoetherian objects”.
add a commentÂ Â
add a commentÂ Â
1
down vote
The finitely presented rings are the compact objects of the category of rings.
That is, those objects $R$ such that the functor $hom(R,)colon mathrm{(Ring)}tomathrm{(Set)}$ preserves filtered colimits.
In fact, this applies to any variety of algebras, e.g., by JiÃ
ÂÃÂ AdÃÂ¡mek, JiÃ
ÂÃÂ RosickÃÂ½, Locally Presentable and Accessible Categories, Corollary 3.13.

Also see qchu.wordpress.com/2015/04/25/compactobjects and replace modules by rings for a proof.
âÂ Ben
22 mins ago
add a commentÂ Â
1
down vote
The finitely presented rings are the compact objects of the category of rings.
That is, those objects $R$ such that the functor $hom(R,)colon mathrm{(Ring)}tomathrm{(Set)}$ preserves filtered colimits.
In fact, this applies to any variety of algebras, e.g., by JiÃ
ÂÃÂ AdÃÂ¡mek, JiÃ
ÂÃÂ RosickÃÂ½, Locally Presentable and Accessible Categories, Corollary 3.13.

Also see qchu.wordpress.com/2015/04/25/compactobjects and replace modules by rings for a proof.
âÂ Ben
22 mins ago
add a commentÂ Â
1
down vote
1
down vote
The finitely presented rings are the compact objects of the category of rings.
That is, those objects $R$ such that the functor $hom(R,)colon mathrm{(Ring)}tomathrm{(Set)}$ preserves filtered colimits.
In fact, this applies to any variety of algebras, e.g., by JiÃ
ÂÃÂ AdÃÂ¡mek, JiÃ
ÂÃÂ RosickÃÂ½, Locally Presentable and Accessible Categories, Corollary 3.13.
The finitely presented rings are the compact objects of the category of rings.
That is, those objects $R$ such that the functor $hom(R,)colon mathrm{(Ring)}tomathrm{(Set)}$ preserves filtered colimits.
In fact, this applies to any variety of algebras, e.g., by JiÃ
ÂÃÂ AdÃÂ¡mek, JiÃ
ÂÃÂ RosickÃÂ½, Locally Presentable and Accessible Categories, Corollary 3.13.

Also see qchu.wordpress.com/2015/04/25/compactobjects and replace modules by rings for a proof.
âÂ Ben
22 mins ago
add a commentÂ Â

Also see qchu.wordpress.com/2015/04/25/compactobjects and replace modules by rings for a proof.
âÂ Ben
22 mins ago
âÂ Ben
22 mins ago
âÂ Ben
22 mins ago
add a commentÂ Â
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