Fields of Fractions and the CRT

Rings of fractions invert the elements that cancellation already treats like units, producing the field of fractions of a domain. The Chinese Remainder Theorem reverses a direct product: when a ring is quotiented by ideals that are pairwise coprime, the quotient factors into a product of simpler quotients. Both constructions are modeled on the integers — $\mathbb{Z}\hookrightarrow \mathbb{Q}$ and $\mathbb{Z}/mn\mathbb{Z}\cong \mathbb{Z}/m\mathbb{Z}\times \mathbb{Z}/n\mathbb{Z}$ — and both hold for general commutative rings.

Inverting cancellable elements

In a commutative ring, a nonzero element $a$ that is not a zero divisor already cancels: $ab=ac$ forces $b=c$. It behaves like a unit without necessarily being one — $2\in \mathbb{Z}$ cancels but has no integer inverse. The construction below manufactures the missing inverses in a larger ring, exactly as the rational numbers supply inverses for the integers.¹

Two restrictions are forced before starting. If $b$ is $0$ or a zero divisor, say $bd=0$ with $d\ne 0$, then allowing $b$ as a denominator collapses the structure: $\frac{d}{1}=\frac{bd}{b}=\frac{0}{b}=0$, so $R$ could not embed. And the product rule needs the set of denominators closed under multiplication, since $bd$ must also be a denominator. The theorem requires these two conditions and nothing more.

The ring of fractions

The construction copies the arithmetic of $\mathbb{Q}$. A fraction is an ordered pair $(r,d)$ with $d\in D$, and two fractions are identified when they cross-multiply equal.

A single rational number has many representations, $\frac{1}{2}=\frac{2}{4}=\frac{3}{6}$, and the relation $\sim$ collapses them to one class. Geometrically the pair $(r,d)$ is a lattice point and its class is the whole ray of proportional points.

The ring $D^{-1}R$ is the smallest one containing $R$ in which the elements of $D$ invert, and smallest is a universal property.

Any ring in which $R$ sits with its $D$-elements inverted must already contain a copy of $D^{-1}R$.¹ The uniqueness of this extension is why $D^{-1}R$ is called the ring of fractions rather than merely a ring of fractions.

The field of fractions

The construction is most useful when $R$ is a domain and every nonzero element is allowed as a denominator.

Since every nonzero element becomes a unit, $D^{-1}R$ is a field. It is the smallest field containing $R$: any field holding a copy of $R$ holds a copy of its field of fractions.¹²

• $\mathbb{Z}$ has field of fractions $\mathbb{Q}$.
• A quadratic integer ring $\mathcal{O}\subset \mathbb{Q}(\sqrt{D})$ has field of fractions the quadratic field $\mathbb{Q}(\sqrt{D})$ itself.
• $2\mathbb{Z}$, a domain-like ring without identity, still has field of fractions $\mathbb{Q}$ — the identity appears in the fractions.
• For any domain $R$, the polynomial ring $R[x]$ has field of fractions the field of rational functions $\frac{p(x)}{q(x)}$, written $F(x)$ when $F$ is a field. This is a recurring base field in field theory.

Restricting $D$ to the powers $\{1,d,d^2,\dots \}$ of a single element gives $R[\frac{1}{d}]$, whose elements look like polynomials in $\frac{1}{d}$. This localization at $d$ inverts one element without inverting all of them, and is the local-to-global tool of commutative algebra.¹

Direct products and comaximal ideals

The direct product $R_1\times \cdots \times R_k$ of rings has componentwise addition and multiplication. A map into a product is a homomorphism exactly when each coordinate map is. The Chinese Remainder Theorem says when a single ring is already such a product, viewed through its ideals. The bridge is the ideal-theoretic form of

In $\mathbb{Z}$ this is exactly coprimality: $m\mathbb{Z}+n\mathbb{Z}=\mathbb{Z}$ holds iff $\gcd (m,n)=1$, since $m\mathbb{Z}+n\mathbb{Z}=\gcd (m,n)\mathbb{Z}$ and the Bézout relation $mx+ny=1$ certifies it.³

The Chinese Remainder Theorem

For $k=2$ the argument is short. The map is a homomorphism because each coordinate is a projection, and its kernel is the set of elements lying in both $A$ and $B$, that is $A\cap B$. Comaximality supplies $x\in A$, $y\in B$ with $x+y=1$; then $x\mapsto (0,1)$ and $y\mapsto (1,0)$, so $r_{2}x+r_{1}y$ hits any target $(r_1+A,r_2+B)$, proving surjectivity. Finally any $c\in A\cap B$ satisfies $c=cx+cy\in AB$, giving $A\cap B\subseteq AB\subseteq A\cap B$.³ The general case follows by induction once $A_1$ and $A_2\cdots A_k$ are shown comaximal.

The named special case is the isomorphism $\mathbb{Z}/mn\mathbb{Z}\cong \mathbb{Z}/m\mathbb{Z}\times \mathbb{Z}/n\mathbb{Z}$ for $\gcd (m,n)=1$: each residue mod $mn$ is determined by its pair of residues mod $m$ and mod $n$, and every pair occurs exactly once.

Reading off the units on both sides shows they correspond, since a unit in a product is a unit in each factor:

Comparing orders recovers the multiplicativity of the Euler function, $\phi (mn)=\phi (m)\phi (n)$ for coprime $m,n$, and iterating over the prime-power factorization $n=p_1^{a_1}\cdots p_k^{a_k}$ gives both $\mathbb{Z}/n\mathbb{Z}\cong {\prod }_i\mathbb{Z}/p_i^{a_i}\mathbb{Z}$ and the product formula for $\phi (n)$.³ This is the ring-theoretic source of the counting facts used in cosets and Euler's theorem and in the classification of finite abelian groups.

Reconstructing from residues

The CRT is constructive: given the residues, the original element is recovered by a fixed formula. For pairwise coprime moduli $n_1,\dots ,n_k$ with product $n$, let $m_i=n/n_i$ be the product of the other moduli, coprime to $n_i$; its inverse $t_i\equiv m_i^{-1}(\mathrm{mod}{n}_i)$ exists and is found by the extended Euclidean algorithm.

Each term $a_i{m}_i{t}_i$ is $\equiv a_i(\mathrm{mod}{n}_i)$ and $\equiv 0$ modulo every other $n_j$, so the sum satisfies all $k$ congruences at once, uniquely modulo $n$.⁴

This is the basis of a residue-number-system representation. A calculation on integers too large for a machine word can be split across several coprime moduli, each fitting in a word; additions and multiplications run independently on each residue, and the CRT reassembles the exact result at the end. Because the residues never interact, the component calculations parallelize with no carries between them.⁴