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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 $i \neq j$, $I_i + I_j = n$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}/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$-subgroups. There are Sylow decompositions in certain other group-theoretic contexts as well, e.g. nilpotent profinite groups. 1 The Chinese Remainder theorem is usually thought of as an isomorphism of rings, not just of cyclic groups. In this regard it has a vast generalization: Theorem (Ideal-theoretic CRT): Let R be a commutative ring, and let$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 = n$. Then$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}/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.