# nLab core of a ring

Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Definition

###### Definition

For $R$ a commutative ring, its 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 commutative ring). That is, it is the smallest regular monomorphism into $R$ in the category CRing.

By the general construction of regular images (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 (prop.).

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

$c R \overset{equ}{\longrightarrow} R \stackrel{\longrightarrow}{\longrightarrow} R \otimes R \,,$

where the top morphism is

$R \simeq \mathbb{Z} \otimes R \overset{e \otimes id}{\longrightarrow} R \otimes R$

and the bottom one is

$R \simeq R \otimes \mathbb{Z} \overset{id \otimes e}{\longrightarrow} R \otimes R \,.$

A ring which is isomorphic to its core is called a solid ring.

###### Remark

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.), 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 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. .

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, see 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 ((infinity,1)-image) of $Spec(E) \to$ Spec(S) (see here).

## Classification

###### Theorem

The following is the complete list of solid rings (def. ) up to isomorphism:

1. The localization of the ring of integers at a set $J$ of prime numbers

$\mathbb{Z}[J^{-1}] \,;$
2. the cyclic rings

$\mathbb{Z}/n\mathbb{Z}$

for $n \in \mathbb{N}$, $n \geq 2$;

3. the product rings

$\mathbb{Z}[J^{-1}] \times \mathbb{Z}/n\mathbb{Z} \,,$

for $n \geq 2$ such that each prime factor of $n$ is contained in the set of primes $J$;

4. the ring cores of product rings

$c(\mathbb{Z}[J^{-1}] \times \underset{p \in K}{\prod} \mathbb{Z}/p^{e(p)}) \,,$

where $K \subset J$ are infinite sets of primes and $e(p)$ are positive natural numbers.

## Properties

###### Proposition

The core of any ring $R$ is solid (def. ):

$c c R \simeq c R \,.$