The idea hit me when I was in my Elliptic Curve Cryptography class. $Z\_n \leftrightarrow Z\_{f\_1} \times Z\_{f\_2} \times ...$ where $f\_1 \times f\_2 \times ... = n$ and $\{f\_1, f\_2, ...\}$ are pairwise coprime. Applications of this Chinese Remainder Theorem not only include computational speedups (in the case of decryption in RSA) but also stronger cryptographic attacks against $Z\_n$ (for example, Pollard Rho factoring exploits this structure). Can we extend these applications into other areas? (Admittedly I don't know many computationaly examples where this could be useful, but can imagine that Mathematica/Maple would find some uses).  
  



So the real question: is this property just a "coincidence" or are there analogs in other groups? If there are, is there some group theory analog that applies equally well to every group? If there are not, what underlying structure in the natural numbers makes this possible?



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://alpha.math.uga.edu/~pete/integral.pdf.](http://alpha.math.uga.edu/%7Epete/integral.pdf.))


One could also think of $\mathbb{Z}/n\mathbb{Z}$ 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.


