Analog to the Chinese Remainder Theorem in groups other than Z_n.

The Chinese Remainder theorem is usually thought of as an isomorphism of rings, not just of cyclic groups. In this regard it has a vast generalization:

Theorem (Ideal-theoretic CRT): Let R be a commutative ring, and let $I_1,ldots,I_n$ be a finite set of ideals in $R$ which are pairwise comaximal: for all $i neq j$, $I_i + I_j = R$. Then $I_1 cap ldots cap I_n = I_1 cdots I_n$ and the natural homomorphism

$R/I_1 cdots I_n = R/I_1 cap ldots cap I_n rightarrow bigoplus_{i=1}^n R/I_i$

is an isomorphism. (See e.g. Theorem 41 on p.31 of http://math.uga.edu/~pete/integral.pdf.)

One could also think of $mathbb{Z}/nmathbb{Z}$ as a $mathbb{Z}$-module, and then the CRT decomposition is a special case of primary decomposition for $R$-modules. In general rings, primary decomposition is somewhat complicated (e.g. it need not be unique), but for finitely generated torsion modules over a PID there is a straightforward analogue.

Finally, thinking about it in terms of groups, CRT has the following generalization: a finite group is nilpotent iff each Sylow $p$-subgroup is normal and $G$ is the direct product of its Sylow $p$-subgroups. There are Sylow decompositions in certain other group-theoretic contexts as well, e.g. nilpotent profinite groups.

Here is a variation on CRT for unit groups in modular arithmetic when the two moduli $m$ and $n$ may not be relatively prime: the natural reduction/diagonal map $(mathbf Z/mnmathbf Z)^times rightarrow (mathbf Z/mmathbf Z)^times times (mathbf Z/nmathbf Z)^times$
need not be an isomorphism, but it always fits into an exact sequence
$$
1 rightarrow K rightarrow (mathbf Z/mnmathbf Z)^times rightarrow (mathbf Z/mmathbf Z)^times times (mathbf Z/nmathbf Z)^times rightarrow (mathbf Z/dmathbf Z)^times rightarrow 1,
$$
where $K$ is all $a bmod mn$ such that $a equiv 1 bmod text{lcm}(m,n)$ and
$d = gcd(m,n)$, with the map to $(mathbf Z/dmathbf Z)^times$ being given by $(u,v) mapsto uv^{-1}bmod d$. (All this is really doing is making the cokernel explicit as a unit group.) Note $K$ has size $mn / text{lcm}(m,n) = gcd(m,n) = d$.

Taking the alternating product of the sizes of these groups, we get
$$
frac{d cdot varphi(m)varphi(n)}{varphi(mn)cdot varphi(d)} = 1 Longrightarrow varphi(mn) = varphi(m)varphi(n)frac{d}{varphi(d)},
$$
which is a formula for $varphi(mn)$ even if $m$ and $n$ are not relatively prime.
Of course this last formula can be derived directly from manipulations with Euler’s formula $varphi(N) = Nprod_{p|N} (1 – 1/p)$ when $N = m$, $n$, and $mn$, but that makes it seem kind of accidental. Is such a formula for $varphi(mn)$ in case $gcd(m,n) not= 1$ of any use at all? I know two applications: a proof that if $mathbf Q(zeta_m) = mathbf Q(zeta_n)$ then $m = n$ or one of $m$ and $n$ is odd and the other number is its double (like $mathbf Q(zeta_5)= mathbf Q(zeta_{10})$ since $-zeta_5$ has order 10) and a proof that $mathbf Q(zeta_m) cap mathbf Q(zeta_n) = mathbf Q(zeta_{(m,n)})$.

What you know as the Chinese remainder theorem for the abelian group $mathbb{Z}/nmathbb{Z}$ (which you probably don’t want to call a “simple group” unless $n$ is prime, as this term has a technical meaning that doesn’t apply to composite $n$) is a special case of a general result in basic ring theory that can be found in any introductory text on algebra (for instance, the text of Hungerford, where the result without requiring the ring to have an identity, so there’s a bizarre extra hypothesis, or Dummit and Foote, where it is stated in its usual form). The result is especially useful in the theory of Dedekind domains, which can be thought of as a generalization of the ring of integers.

I have a feeling people are going to cite this question as being somewhat inappropriate for the site, as it isn’t research level, but I figured I’d attempt to steer you in the right direction anyway.

Leave a Comment