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\begin{document}

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\section*{core of a ring}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{classification}{Classification}\dotfill \pageref*{classification} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{CoreOfARing}\hypertarget{CoreOfARing}{}
For $R$ a [[commutative ring]], its \textbf{[[core of a ring|core]]} $c R$ is the [[regular image]] of the unique [[ring]] [[homomorphism]] $\mathbb{Z} \overset{e}{\longrightarrow} R$ (note that $\mathbb{Z}$ is the [[initial object|initial]] commutative ring). That is, it is the smallest [[regular monomorphism]] into $R$ in the category [[CRing]].

By the general construction of regular images (\href{image#AsEqualizer}{here}), this can be computed as the [[equalizer]] of the two inclusions from $R$ into the [[pushout]] $R\sqcup_{\mathbb{Z}} R$. Since $\mathbb{Z}$ is initial, this is just the [[coproduct]] $R\sqcup R$ in $CRing$, which is the [[tensor product of abelian groups]] $R\otimes R$ equipped with its canonically induced commutative ring structure (\href{CRing#CoproductIsTensorProduct}{prop.}).

Thus the most explicit definition of $c R$ is that it is the [[equalizer]] in

\begin{displaymath}
c R 
    \overset{equ}{\longrightarrow}
  R 
    \stackrel{\longrightarrow}{\longrightarrow}
  R \otimes R
  \,,
\end{displaymath}
where the top morphism is

\begin{displaymath}
R \simeq \mathbb{Z} \otimes R \overset{e \otimes id}{\longrightarrow} R \otimes R
\end{displaymath}
and the bottom one is

\begin{displaymath}
R \simeq R \otimes \mathbb{Z}  \overset{id \otimes e}{\longrightarrow} R \otimes R
  \,.
\end{displaymath}
A ring which is [[isomorphism|isomorphic]] to its core is called a \textbf{solid ring}.

\end{defn}
(\hyperlink{BousfieldKan72}{Bousfield-Kan 72, \S{}1, def. 2.1}, \hyperlink{Bousfield79}{Bousfield 79, 6.4})

\begin{remark}
\label{DualInterpretation}\hypertarget{DualInterpretation}{}
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

\begin{displaymath}
Spec(R) \longrightarrow Spec(Z)
\end{displaymath}
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 (\href{CRing#CoproductIsTensorProduct}{prop.}), this is the dually the [[Cartesian product]] in $CRing^{op}$ and hence

\begin{displaymath}
Spec(R \otimes R) \simeq Spec(R) \times Spec(R)
\end{displaymath}
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 $CRing^{op}$

\begin{displaymath}
\itexarray{
    Spec(R) \times Spec(R) \times Spec(R)
    \\
    \downarrow
    \\
    Spec(R) \times Spec(R)
    \\
    {}^{\mathllap{s}}\downarrow \uparrow \downarrow^{\mathrlap{t}}
    \\
    Spec(R)
  }
  \,.
\end{displaymath}
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

\begin{displaymath}
Spec(R) \times Spec(R)
   \underoverset
     {\underset{s}{\longrightarrow}}
     {\overset{t}{\longrightarrow}}
     {\phantom{AA}}
   Spec(R)
     \overset{coeq}{\longrightarrow}
   Spec(c R)
  \,,
\end{displaymath}
also called the \emph{[[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 (\hyperlink{Bousfield79}{Bousfield 79, theorems 6.5, 6.6}, see \href{Adams+spectral+sequence#ConvergenceStatements}{here}), because the $E$-Adams spectral sequence computes [[E-nilpotent completion]] which is essentially the analog in [[higher algebra]] of the above story: namely the coimage (\href{infinity-image#ViaColimitOfCechNerve}{(infinity,1)-image}) of $Spec(E) \to$ [[Spec(S)]] (see \href{Adams+spectral+sequence#DefinitionInHigherAlgebra}{here}).

\end{remark}
\hypertarget{classification}{}\subsection*{{Classification}}\label{classification}

\begin{theorem}
\label{}\hypertarget{}{}
The following is the complete list of solid rings (def. \ref{CoreOfARing}) up to [[isomorphism]]:

\begin{enumerate}%
\item The [[localization of a ring|localization]] of the ring of [[integers]] at a set $J$ of [[prime numbers]]

\begin{displaymath}
\mathbb{Z}[J^{-1}]
  \,;
\end{displaymath}

\item the [[cyclic rings]]

\begin{displaymath}
\mathbb{Z}/n\mathbb{Z}
\end{displaymath}
for $n \in \mathbb{N}$, $n \geq 2$;


\item the [[Cartesian product|product]] rings

\begin{displaymath}
\mathbb{Z}[J^{-1}]
    \times
  \mathbb{Z}/n\mathbb{Z}
  \,,
\end{displaymath}
for $n \geq 2$ such that each [[prime factor]] of $n$ is contained in the set of primes $J$;


\item the ring cores of product rings

\begin{displaymath}
c(\mathbb{Z}[J^{-1}] \times \underset{p \in K}{\prod} \mathbb{Z}/p^{e(p)})
  \,,
\end{displaymath}
where $K \subset J$ are infinite sets of primes and $e(p)$ are positive natural numbers.



\end{enumerate}
\end{theorem}
(\hyperlink{BousfieldKan72}{Bousfield-Kan 72, prop. 3.5}, \hyperlink{Bousfield79}{Bousfield 79, p. 276})

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{prop}
\label{}\hypertarget{}{}
The core of any ring $R$ is solid (def. \ref{CoreOfARing}):

\begin{displaymath}
c c R \simeq c R
  \,.
\end{displaymath}
\end{prop}
(\hyperlink{BousfieldKan72}{Bousfield-Kan 72, prop. 2.2})

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[nilpotent completion of spectra]]


\item [[Adams spectral sequence]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Aldridge Bousfield]], [[Daniel Kan]], \emph{The core of a ring}, Journal of Pure and Applied Algebra, Volume 2, Issue 1, April 1972, Pages 73-81 (\href{http://www.sciencedirect.com/science/article/pii/0022404972900230}{link})


\item [[Aldridge Bousfield]], \emph{The localization of spectra with respect to homology}, Topology 18 (1979), no. 4, 257--281. (\href{http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/bousfield-topology-1979.pdf}{pdf})



\end{itemize}
[[!redirects cores of rings]]

[[!redirects solid ring]] [[!redirects solid rings]]



\end{document}