Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

Discussion Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

nLab > Latest Changes: core of a ring

Bottom of Page

1 to 17 of 17

- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeJul 15th 2016
- PermaLink
Author: Urs
Format: MarkdownItexgave _[[core of a ring]]_ some minimum content

gave core of a ring some minimum content
- CommentRowNumber2.
- CommentAuthorTodd_Trimble
- CommentTimeJul 15th 2016
- PermaLink
Author: Todd_Trimble
Format: MarkdownItexI added the word "commutative", and added a link to the Bousfield-Kan paper.

I added the word “commutative”, and added a link to the Bousfield-Kan paper.
- CommentRowNumber3.
- CommentAuthorUrs
- CommentTimeJul 15th 2016
- PermaLink
Author: Urs
Format: MarkdownItexAh, thanks for catching that.

Ah, thanks for catching that.
- CommentRowNumber4.
- CommentAuthorMike Shulman
- CommentTimeJul 15th 2016
- PermaLink
Author: Mike Shulman
Format: MarkdownItexI added a remark about [[regular image]]s.

I added a remark about regular images.
- CommentRowNumber5.
- CommentAuthorUrs
- CommentTimeJul 15th 2016
- PermaLink
Author: Urs
Format: MarkdownItexThanks! I hadn't realized that relation.

Thanks! I hadn’t realized that relation.
- CommentRowNumber6.
- CommentAuthorMike Shulman
- CommentTimeJul 15th 2016
- PermaLink
Author: Mike Shulman
Format: MarkdownItexIt makes me wonder, what are the regular monos in general in $CRing$?

It makes me wonder, what are the regular monos in general in $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>CRing</mi></mrow><annotation encoding="application/x-tex">CRing</annotation></semantics></math>$ ?
- CommentRowNumber7.
- CommentAuthorUrs
- CommentTimeJul 16th 2016
- PermaLink
Author: Urs
Format: MarkdownItexI have added a remark [here](https://ncatlab.org/nlab/show/core+of+a+ring#DualInterpretation) on the dual interpretation. (my battery is dying right now, possibly there are typos left): We may think of the [[opposite category]] $CRing^{op}$ as that of affine [[arithmetic schemes]]. Here for $R \in CRing$ we write $Spec(R)$ for the same object, but regarded in $CRing^{op}$. So the [[initial object]] $\mathbb{Z}$ in [[CRing]] becomes the [[terminal object]] [[Spec(Z)]] in $CRing^{op}$, and so for every $R$ there is a unique morphism $$ Spec(R) \longrightarrow Spec(Z) $$ in $CRing^{op}$, exhibiting every affine [[arithmetic scheme]] $Spec(R)$ as equipped with a map to the base scheme [[Spec(Z)]]. Since the [[coproduct]] in [[CRing]] is the [[tensor product]] of rings ([prop.](CRing#CoproductIsTensorProduct)), this is the dually the [[Cartesian product]] in $CRing^{op}$ and hence $$ Spec(R \otimes R) \simeq Spec(R) \times Spec(R) $$ exhibits $R \otimes R$ as the ring of functions on $Spec(R) \times Spec(R)$. Hence the terminal morphism $Spec(R) \to Spec(\mathbb{Z})$ induced the corresponding [[Cech groupoid]] [[internal groupoid|internal]] to $CRing^{op}$ $$ \array{ Spec(R) \times Spec(R) \times Spec(R) \\ \downarrow \\ Spec(R) \times Spec(R) \\ {}^{\mathllap{s}}\downarrow \uparrow \downarrow^{\mathrlap{t}} \\ Spec(R) } \,. $$ This exhibits $R \otimes R$ (the ring of functions on the scheme of morphisms of the Cech groupoid) as a [[commutative Hopf algebroid]] over $R$. Moreover, the arithmetic scheme of [[isomorphism classes]] of this groupoid is the [[coequalizer]] of the [[source]] and [[target]] morphisms $$ Spec(R) \times Spec(R) \underoverset {\underset{s}{\longrightarrow}} {\overset{t}{\longrightarrow}} {\phantom{AA}} Spec(R) \overset{coeq}{\longrightarrow} Spec(c R) \,, $$ also called the _[[coimage]]_ of $Spec(R) \to Spec(\mathbb{Z})$. Since [[limits]] in the [[opposite category]] $CRing^{op}$ are equivaletly colimits in $CRing$, this means that the ring of functions on the scheme of isomorphism classes of the Cech groupoid is precisely the core $c R$ or $R$ according to def. \ref{CoreOfARing}. This is morally the reason why for $E$ a [[homotopy commutative ring spectrum]] then the core $c \pi_0(E)$ of its underlying ordinary ring in degree 0 controls what the $E$-[[Adams spectral sequence]] converges to ([Bousfield 79, theorems 6.5, 6.6](#Bousfield79), see [here](Adams+spectral+sequence#ConvergenceStatements)), because the $E$-Adams spectral sequence computes [[E-nilpotent completion]] which is essentially the analog in [[higher alegbra]] of the above story: namely the coimage ([[(infinity,1)-image]]) of $Spec(E) \to $ [[Spec(S)]] (see [here](Adams+spectral+sequence#DefinitionInHigherAlgebra)).

I have added a remark here on the dual interpretation. (my battery is dying right now, possibly there are typos left):

We may think of the opposite category $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>CRing</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">CRing^{op}</annotation></semantics></math>$ as that of affine arithmetic schemes. Here for $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mo>∈</mo><mi>CRing</mi></mrow><annotation encoding="application/x-tex">R \in CRing</annotation></semantics></math>$ we write $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(R)</annotation></semantics></math>$ for the same object, but regarded in $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>CRing</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">CRing^{op}</annotation></semantics></math>$ .

So the initial object $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>$ in CRing becomes the terminal object Spec(Z) in $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>CRing</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">CRing^{op}</annotation></semantics></math>$ , and so for every $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>$ there is a unique morphism
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Spec(R) \longrightarrow Spec(Z) </annotation></semantics></math>$
in $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>CRing</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">CRing^{op}</annotation></semantics></math>$ , exhibiting every affine arithmetic scheme $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(R)</annotation></semantics></math>$ as equipped with a map to the base scheme Spec(Z).

Since the coproduct in CRing is the tensor product of rings (prop.), this is the dually the Cartesian product in $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>CRing</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">CRing^{op}</annotation></semantics></math>$ and hence
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo>⊗</mo><mi>R</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>×</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Spec(R \otimes R) \simeq Spec(R) \times Spec(R) </annotation></semantics></math>$
exhibits $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mo>⊗</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">R \otimes R</annotation></semantics></math>$ as the ring of functions on $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>×</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(R) \times Spec(R)</annotation></semantics></math>$ .

Hence the terminal morphism $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(R) \to Spec(\mathbb{Z})</annotation></semantics></math>$ induced the corresponding Cech groupoid internal to $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>CRing</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">CRing^{op}</annotation></semantics></math>$
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>×</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>×</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>×</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow/> <mpadded width="0px" lspace="-100%width"><mi>s</mi></mpadded></msup><mo stretchy="false">↓</mo><mo stretchy="false">↑</mo><msup><mo stretchy="false">↓</mo> <mpadded width="0px"><mi>t</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="0.16667em"/><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ Spec(R) \times Spec(R) \times Spec(R) \\ \downarrow \\ Spec(R) \times Spec(R) \\ {}^{\mathllap{s}}\downarrow \uparrow \downarrow^{\mathrlap{t}} \\ Spec(R) } \,. </annotation></semantics></math>$
This exhibits $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mo>⊗</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">R \otimes R</annotation></semantics></math>$ (the ring of functions on the scheme of morphisms of the Cech groupoid) as a commutative Hopf algebroid over $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>$ .

Moreover, the arithmetic scheme of isomorphism classes of this groupoid is the coequalizer of the source and target morphisms
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>×</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><munderover><mphantom><mi>AA</mi></mphantom><munder><mo>⟶</mo><mi>s</mi></munder><mover><mo>⟶</mo><mi>t</mi></mover></munderover><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mi>coeq</mi></mover><mi>Spec</mi><mo stretchy="false">(</mo><mi>c</mi><mi>R</mi><mo stretchy="false">)</mo><mspace width="0.16667em"/><mo>,</mo></mrow><annotation encoding="application/x-tex"> Spec(R) \times Spec(R) \underoverset {\underset{s}{\longrightarrow}} {\overset{t}{\longrightarrow}} {\phantom{AA}} Spec(R) \overset{coeq}{\longrightarrow} Spec(c R) \,, </annotation></semantics></math>$
also called the coimage of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(R) \to Spec(\mathbb{Z})</annotation></semantics></math>$ . Since limits in the opposite category $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>CRing</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">CRing^{op}</annotation></semantics></math>$ are equivaletly colimits in $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>CRing</mi></mrow><annotation encoding="application/x-tex">CRing</annotation></semantics></math>$ , this means that the ring of functions on the scheme of isomorphism classes of the Cech groupoid is precisely the core $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">c R</annotation></semantics></math>$ or $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>$ according to def. \ref{CoreOfARing}.

This is morally the reason why for $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>$ a homotopy commutative ring spectrum then the core $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c \pi_0(E)</annotation></semantics></math>$ of its underlying ordinary ring in degree 0 controls what the $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>$ -Adams spectral sequence converges to (Bousfield 79, theorems 6.5, 6.6, see here), because the $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>$ -Adams spectral sequence computes E-nilpotent completion which is essentially the analog in higher alegbra of the above story: namely the coimage ((infinity,1)-image) of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>→</mo></mrow><annotation encoding="application/x-tex">Spec(E) \to </annotation></semantics></math>$ Spec(S) (see here).
- CommentRowNumber8.
- CommentAuthorUrs
- CommentTimeJul 18th 2016
- (edited Jul 18th 2016)
- PermaLink
Author: Urs
Format: MarkdownItexIs there established terminology for the generalization of the concept to [[commutative Hopf algebroids]]? I.e. for $$ A \underoverset {\underset{\eta_R}{\longrightarrow}} {\overset{\eta_L}{\longrightarrow}} {\phantom{AA}} \Gamma $$ the left and right unit maps of a commutative Hopf algebroid $\Gamma$ over $A$, is there established terminology for their equalizer? (So that the core of a ring $R$ would be the special case for the commutative Hopf algebroid $R \otimes R$ over $R$...)

Is there established terminology for the generalization of the concept to commutative Hopf algebroids?

I.e. for
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>A</mi><munderover><mphantom><mi>AA</mi></mphantom><munder><mo>⟶</mo><mrow><msub><mi>η</mi> <mi>R</mi></msub></mrow></munder><mover><mo>⟶</mo><mrow><msub><mi>η</mi> <mi>L</mi></msub></mrow></mover></munderover><mi>Γ</mi></mrow><annotation encoding="application/x-tex"> A \underoverset {\underset{\eta_R}{\longrightarrow}} {\overset{\eta_L}{\longrightarrow}} {\phantom{AA}} \Gamma </annotation></semantics></math>$
the left and right unit maps of a commutative Hopf algebroid $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math>$ over $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>$ , is there established terminology for their equalizer?

(So that the core of a ring $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>$ would be the special case for the commutative Hopf algebroid $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mo>⊗</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">R \otimes R</annotation></semantics></math>$ over $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>$ …)
- CommentRowNumber9.
- CommentAuthorUrs
- CommentTimeJul 17th 2021
- PermaLink
Author: Urs
Format: MarkdownItexLinked [the remark](https://ncatlab.org/nlab/show/core+of+a+ring#DualInterpretation) on the geometric interpretation to *[[duality between algebra and geometry]]*. <a href="https://ncatlab.org/nlab/revision/diff/core+of+a+ring/9">diff</a>, <a href="https://ncatlab.org/nlab/revision/core+of+a+ring/9">v9</a>, <a href="https://ncatlab.org/nlab/show/core+of+a+ring">current</a>

Linked the remark on the geometric interpretation to duality between algebra and geometry.

diff, v9, current
- CommentRowNumber10.
- CommentAuthorUrs
- CommentTimeJul 17th 2021
- (edited Jul 17th 2021)
- PermaLink
Author: Urs
Format: MarkdownItexI have slightly reworked the Definition-section for readability: Gave the definition as a subset, stated up-front, then followed by a remark which expands on the category-theoretic formulation as an equalizer and regular image. Also added the statement that a commutative ring is solid iff its multiplication is an isomorphism. <a href="https://ncatlab.org/nlab/revision/diff/core+of+a+ring/10">diff</a>, <a href="https://ncatlab.org/nlab/revision/core+of+a+ring/10">v10</a>, <a href="https://ncatlab.org/nlab/show/core+of+a+ring">current</a>

I have slightly reworked the Definition-section for readability:

Gave the definition as a subset, stated up-front, then followed by a remark which expands on the category-theoretic formulation as an equalizer and regular image.

Also added the statement that a commutative ring is solid iff its multiplication is an isomorphism.

diff, v10, current
- CommentRowNumber11.
- CommentAuthorUrs
- CommentTimeJul 17th 2021
- PermaLink
Author: Urs
Format: MarkdownItexAdded these references: *** The concept re-appears under the name "T-rings" in * R. A. Bowshell and P. Schultz, *Unital rings whose additive endomorphisms commute*, Mathematische Annalen volume 228, pages 197–214 (1977) ([doi10.1007/BF01420290](https://doi.org/10.1007/BF01420290)) and under the name "$\mathbb{Z}$-epimorphs" in: * Warren Dicks, W. Stephenson, *Epimorphs and Dominions of Dedekind Domains*, Journal of the London Mathematical Society, Volume s2-29, Issue 2, April 1984, Pages 224–228 ([doi:10.1112/jlms/s2-29.2.224](https://doi.org/10.1112/jlms/s2-29.2.224)) Generalization to [[monoids in monoidal categories]]: * [[Javier J. Gutiérrez]], *On solid and rigid monoids in monoidal categories*, Applied Categorical Structures 23, no. 2 (2015), 575-589 ([arXiv:1303.5265](https://arxiv.org/abs/1303.5265)) *** <a href="https://ncatlab.org/nlab/revision/diff/core+of+a+ring/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/core+of+a+ring/11">v11</a>, <a href="https://ncatlab.org/nlab/show/core+of+a+ring">current</a>
Added these references:

The concept re-appears under the name “T-rings” in
- R. A. Bowshell and P. Schultz, Unital rings whose additive endomorphisms commute, Mathematische Annalen volume 228, pages 197–214 (1977) (doi10.1007/BF01420290)
and under the name “ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>$ -epimorphs” in:
- Warren Dicks, W. Stephenson, Epimorphs and Dominions of Dedekind Domains, Journal of the London Mathematical Society, Volume s2-29, Issue 2, April 1984, Pages 224–228 (doi:10.1112/jlms/s2-29.2.224)
Generalization to monoids in monoidal categories:
- Javier J. Gutiérrez, On solid and rigid monoids in monoidal categories, Applied Categorical Structures 23, no. 2 (2015), 575-589 (arXiv:1303.5265)
diff, v11, current
- CommentRowNumber12.
- CommentAuthorUrs
- CommentTimeJul 17th 2021
- (edited Jul 17th 2021)
- PermaLink
Author: Urs
Format: MarkdownItexFor what it's worth, I have made more explicit ([here](https://ncatlab.org/nlab/show/core+of+a+ring#RationalNumbersFormASolidRing)) why the rationals are solid, but no other char=0 field is. <a href="https://ncatlab.org/nlab/revision/diff/core+of+a+ring/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/core+of+a+ring/11">v11</a>, <a href="https://ncatlab.org/nlab/show/core+of+a+ring">current</a>

For what it’s worth, I have made more explicit (here) why the rationals are solid, but no other char=0 field is.

diff, v11, current
- CommentRowNumber13.
- CommentAuthorUrs
- CommentTimeJul 17th 2021
- PermaLink
Author: Urs
Format: MarkdownItexFor what it's worth, I have spelled out a proof ([here](https://ncatlab.org/nlab/show/core+of+a+ring#SolidityMeansThatMultiplicationIsAnIsomorphism)) that ring is solid iff its multiplication is an isomorphism. <a href="https://ncatlab.org/nlab/revision/diff/core+of+a+ring/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/core+of+a+ring/11">v11</a>, <a href="https://ncatlab.org/nlab/show/core+of+a+ring">current</a>

For what it’s worth, I have spelled out a proof (here) that ring is solid iff its multiplication is an isomorphism.

diff, v11, current
- CommentRowNumber14.
- CommentAuthorJohn Baez
- CommentTimeSep 18th 2021
- PermaLink
Author: John Baez
Format: MarkdownItexAdded "idea" of the core of a ring. <a href="https://ncatlab.org/nlab/revision/diff/core+of+a+ring/14">diff</a>, <a href="https://ncatlab.org/nlab/revision/core+of+a+ring/14">v14</a>, <a href="https://ncatlab.org/nlab/show/core+of+a+ring">current</a>

Added “idea” of the core of a ring.

diff, v14, current
- CommentRowNumber15.
- CommentAuthorJohn Baez
- CommentTimeSep 18th 2021
- PermaLink
Author: John Baez
Format: MarkdownItexAdded an "idea" of the core. <a href="https://ncatlab.org/nlab/revision/diff/core+of+a+ring/14">diff</a>, <a href="https://ncatlab.org/nlab/revision/core+of+a+ring/14">v14</a>, <a href="https://ncatlab.org/nlab/show/core+of+a+ring">current</a>

Added an “idea” of the core.

diff, v14, current
- CommentRowNumber16.
- CommentAuthorDELETED_USER_2018
- CommentTimeSep 18th 2021
- (edited Apr 11th 2023)
- PermaLink
Author: DELETED_USER_2018
Format: MarkdownItex[deleted]

[deleted]
- CommentRowNumber17.
- CommentAuthorUrs
- CommentTimeSep 23rd 2021
- PermaLink
Author: Urs
Format: MarkdownItexcross-linked the remark on $\mathbb{R}$ not being solid ([here](https://ncatlab.org/nlab/show/core+of+a+ring#NoOtherCharZeroFieldIsSolid)) with *[[real homotopy theory]]* <a href="https://ncatlab.org/nlab/revision/diff/core+of+a+ring/17">diff</a>, <a href="https://ncatlab.org/nlab/revision/core+of+a+ring/17">v17</a>, <a href="https://ncatlab.org/nlab/show/core+of+a+ring">current</a>

cross-linked the remark on $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>$ not being solid (here) with real homotopy theory

diff, v17, current

1 to 17 of 17

nForum

Discussion Feed

Not signed in

Discussion Tag Cloud

nLab > Latest Changes: core of a ring