Categorical characterizations of ring properties

Clash Royale CLAN TAG#URR8PPP

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.

3

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.

ring-theory category-theory definition big-list

edited 19 mins ago

Hans Stricker

4,91113881

4,91113881

active

oldest

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 co-noetherian objects”.

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ÂÃÂ­ AdÃÂ¡mek, JiÃÂÃÂ­ RosickÃÂ½, Locally Presentable and Accessible Categories, Corollary 3.13.

• Also see qchu.wordpress.com/2015/04/25/compact-objects and replace modules by rings for a proof.
âÂ Ben
22 mins ago

active

oldest

active

oldest

active

oldest

active

oldest

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 co-noetherian 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 co-noetherian 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 co-noetherian 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 co-noetherian objects”.

Arnaud D.

15k52242

15k52242

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ÂÃÂ­ AdÃÂ¡mek, JiÃÂÃÂ­ RosickÃÂ½, Locally Presentable and Accessible Categories, Corollary 3.13.

• Also see qchu.wordpress.com/2015/04/25/compact-objects and replace modules by rings for a proof.
âÂ Ben
22 mins ago

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ÂÃÂ­ AdÃÂ¡mek, JiÃÂÃÂ­ RosickÃÂ½, Locally Presentable and Accessible Categories, Corollary 3.13.

• Also see qchu.wordpress.com/2015/04/25/compact-objects and replace modules by rings for a proof.
âÂ Ben
22 mins ago

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ÂÃÂ­ AdÃÂ¡mek, JiÃÂÃÂ­ RosickÃÂ½, 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ÂÃÂ­ AdÃÂ¡mek, JiÃÂÃÂ­ RosickÃÂ½, Locally Presentable and Accessible Categories, Corollary 3.13.

Ben

4,03421332

4,03421332

• Also see qchu.wordpress.com/2015/04/25/compact-objects and replace modules by rings for a proof.
âÂ Ben
22 mins ago

• Also see qchu.wordpress.com/2015/04/25/compact-objects and replace modules by rings for a proof.
âÂ Ben
22 mins ago

Also see qchu.wordpress.com/2015/04/25/compact-objects and replace modules by rings for a proof.
âÂ Ben
22 mins ago

Also see qchu.wordpress.com/2015/04/25/compact-objects and replace modules by rings for a proof.
âÂ Ben
22 mins ago

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