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  "body": "\n## Summary\n\nA subdirect product of a family of [[algebras|QJGPT2W0FT]] is a [[subalgebra|7CHG6SD365]] of their [[direct product|WJ73XYBNE4]] that still projects onto each factor.\nIt can be much smaller than the full product, yet it remembers each factor completely.\nBy [[Birkhoff's subdirect representation theorem|CZF5696TPY]], every algebra is isomorphic to a subdirect product of [[subdirectly irreducible algebras|DASF939439]].\n\n## Explanations\n\n### L1 human\n\nThe full product of two algebras contains every pair of elements.\nA subdirect product keeps only some pairs, but enough so that every element of each factor still appears in some kept pair.\nFor example, the diagonal $\\set{(x, x)}$ in $\\mathbf{A} \\times \\mathbf{A}$ is a subdirect product of two copies of $\\mathbf{A}$.\nThe integers embed in the product of all $\\Z/p\\Z$ for primes $p$ by sending $n$ to its list of remainders, and this is a subdirect product too.\n",
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