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core of a ring

Definition

Core of a ring

Definition

(core of a ring)
For $R$ a unital commutative ring, its core is the following sub-ring of the tensor product of abelian groups $R \otimes_{{}_{\mathbb{Z}}} R$:

$$c R \;\coloneqq\; \Big\{ \, r \in R \,\left\vert\, 1 \otimes r \,=\, r \otimes 1 \;\in\; R \otimes_{{}_{\mathbb{Z}}} R \right. \, \Big\}$$

(Bousfield & Kan 1972, Sec. 1)

Remark

(core as equalizer and as regular image)
In category-theoretic terminology, Def. describes the equalizer (Bousfield 1979, 6.4)

$$c R \xrightarrow{\;equ\;} R \overset{\phantom{AAAAAA}}\rightrightarrows R \otimes_{{}_{\mathbb{Z}}} R \,,$$

where the top morphism is

$$R \;\simeq\; \mathbb{Z} \otimes R \xrightarrow{e \otimes id} R \otimes R$$

and the bottom one is

$$R \;\simeq\; R \otimes \mathbb{Z} \xrightarrow{\;id \otimes e\;} R \otimes R \,,$$

with

(1)$$\array{ \mathbb{Z} &\xrightarrow{ \;e\; } & R \\ 1 &\mapsto& 1 }$$

denoting the unique ring homomorphism form the commutative ring of integers, which is the initial object in CommutativeRings.

Observing (by this Prop.) that the tensor product of abelian groups $R \otimes_{{}_{\mathbb{Z}}} R$ equipped with its canonically induced commutative ring structure is the coproduct in CommutativeRings

$$R \otimes_{{}_{\mathbb{Z}}} R \;\simeq\; R \sqcup R$$

this means equivalently that the core is the equalizer of the two coprojections into the coproduct:

$$c R \xrightarrow{\;equ\;} R \overset{\phantom{AAAAAA}}\rightrightarrows R \sqcup R \;\simeq\; R \underset{\mathbb{Z}}{\coprod} R \,,$$

hence – since $\mathbb{Z}$ is the initial object in CommutativeRings – into the cofiber coproduct of $\mathbb{Z} \xrightarrow{e} R$ (1) with itself.

In this form, the core is manifestly (here) the regular image of the initial morphism $\mathbb{Z} \xrightarrow{e} R$:

$$c R \;\simeq\; Im_{reg} \left( \mathbb{Z} \xrightarrow{e} R \right) \xhookrightarrow{\phantom{AA}} R \,,$$

hence the smallest regular monomorphism into $R$ in the category of CommutativeRings.

Remark

(geometric interpretation)
By duality between algebra and geometry, we may think of the opposite category $CommutativeRings^{op}$ 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

$$Spec(R) \longrightarrow Spec(\mathbb{Z})$$

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 $Spec(R) \to Spec(\mathbb{Z})$ induces 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).

Solid rings

Definition

(solid rings)
A commutative ring $R$ which is isomorphic to its core (Def. ), $R \,\simeq\, c R$ is called a solid ring.

(Bousfield-Kan 72, §1, def. 2.1, Bousfield 79, 6.4)

Proposition

(solidity means that multiplication is isomorphism)
A commutative ring $R$ is solid (Def. ) iff its multiplication morphism is an isomorphism:

$$R\; \text{is solid} \;\;\;\;\;\;\;\;\; \Leftrightarrow \;\;\;\;\;\;\;\;\; R \otimes_{{}_{\mathbb{Z}}} R \underoverset {\simeq} {\;(-) \cdot (-)\;} {\longrightarrow} R \,.$$

(Bousfield & Kan 1972, 2.4 – this is called a T-ring in Bowshell & Schultz 1977, Def. 1.6)

Proof

In one direction, assume that multiplication is an isomorphism. Since both $r \mapsto 1 \otimes r$ and $r \mapsto r \otimes 1$ are right inverses

and since right inverses of isomorphisms are unique (this Prop.), it follows that these two morphisms on the left are in fact equal, and therefore that their equalizer, hence the core of $R$ (by Rem. ), is $R$.

In the other direction, assume that $1 \otimes r \,=\, r \otimes 1$ for all $r \in R$. Then

$$\begin{aligned} r_1 \,\otimes\, r_2 & \;=\; \big( r_1 \,\otimes\, 1 \big) \cdot \big( 1 \otimes r_2 \big) \\ & \;=\; \big( 1 \,\otimes\, r_1 \big) \cdot \big( 1 \otimes r_2 \big) \\ & \;=\; 1 \,\otimes\, \big( r_1 \cdot r_2 \big) \,, \end{aligned}$$

and therefore $(-)\cdot(-)$ is a bijection, hence an isomorphism of rings.

Properties

Proposition

(cores are solid)
The core (Def. ) of any ring $R$ is solid (Def. ):

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

(Bousfield-Kan 72, prop. 2.2)

Examples

Theorem

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

1. The localization of the ring of integers at a subset $Pr$ of prime numbers

   $$\mathbb{Z}\big[Pr^{-1}\big] \,;$$

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.

(Bousfield-Kan 72, prop. 3.5, Bousfield 79, p. 276)

In particular:

Example

(rational numbers form a solid ring)
The ring $\mathbb{Q}$ of rational numbers is a solid ring.

Proof

Since every rational number may be written as

$$r \,=\,q / p \,=\, \tfrac{1}{p} \cdot q \,\in\, \mathbb{Q}$$

for some $p, q \,\in\, \mathbb{Z}$, we have

$$\begin{aligned} 1 \otimes r & \;=\; 1 \,\otimes\, \big( \tfrac{1}{p} \cdot q \big) \\ & \;=\; \big( \tfrac{1}{p} \cdot p \big) \,\otimes\, \big( \tfrac{1}{p} \cdot q \big) \\ & \;=\; \tfrac{1}{p} \,\otimes\, \big( p \cdot \tfrac{1}{p} \cdot q \big) \\ & \;=\; \tfrac{1}{p} \,\otimes\, \big( q \cdot 1 \big) \\ & \;=\; \big( \tfrac{1}{p} \cdot q \big) \,\otimes\, 1 \\ & \;=\; r \,\otimes\, 1 \;\;\;\;\;\; \in \; \mathbb{Q} \otimes_{{}_{\mathbb{Z}}} \mathbb{Q} \,. \end{aligned}$$

Here the third and the fifth line use the equivalence relation defining the tensor product of abelian groups for the case of rings:

$$r_1 \,\otimes\, \big( n \cdot r_2 \big) \,=\, \big( r_1 \,\cdot\, n \big) \,\otimes\, r_2 \;\;\; \in \; \mathbb{Q} \otimes_{{}_{\mathbb{Z}}} \mathbb{Q} \,, \;\;\;\; \text{for}\; n \,\in\, \mathbb{Z} \,.$$

Remark

(no other field of characteristic zero is a solid ring)
The same argument shows that no other field of characteristic zero $k$ is a solid ring, since, being a proper superset $\mathbb{K} \supset \mathbb{Q}$, it contains elements $k \in \mathbb{K}$ for which there are no pairs of integers $q$, $p$ such that $p \cdot k \,=\, q$. In particular, the real numbers are not a solid ring.

This is, ultimately, the reason why the derived PL de Rham-Quillen adjunction is idempotent only over $k = \mathbb{Q}$ (where it models rationalization of homotopy types, see at fundamental theorem of dg-algebraic rational homotopy theory).

Related concepts

• nilpotent completion of spectra

• Adams spectral sequence

References

The original articles:

• Aldridge Bousfield, Daniel Kan, The core of a ring, Journal of Pure and Applied Algebra, Volume 2, Issue 1, April 1972, Pages 73-81 (doi:10.1016/0022-4049(72)90023-0)

• Aldridge Bousfield, The localization of spectra with respect to homology, Topology 18 (1979), no. 4, 257–281. (pdf)

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 “$\mathbb{Z}$-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)

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)