Category:Chinese Remainder Theorem (Commutative Algebra)
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This category contains pages concerning Chinese Remainder Theorem (Commutative Algebra):
Let $A$ be a commutative and unitary ring.
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Let $I_1, \ldots, I_n$ for some $n \ge 1$ be ideals of $A$.
Then the ring homomorphism $\phi: A \to A / I_1 \times \cdots \times A / I_n$ defined as:
- $\map \phi x = \tuple {x + I_1, \ldots, x + I_n}$
has the kernel $\ds I := \bigcap_{i \mathop = 1}^n I_i$, and is surjective if and only if the ideals are pairwise coprime, that is:
- $\forall i \ne j: I_i + I_j = A$
Hence in that case, it induces an ring isomorphism:
- $A / I \to A / I_1 \times \cdots \times A / I_n$
through the First Isomorphism Theorem.
Pages in category "Chinese Remainder Theorem (Commutative Algebra)"
The following 3 pages are in this category, out of 3 total.