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  "body": "\n## Definition\n\nLet $(\\mathbf{A}_i)_{i \\in I}$ be a family of [[algebras|QJGPT2W0FT]] of the same signature, and let $\\pi_j$ denote the projection of the [[direct product|WJ73XYBNE4]] $\\prod_{i \\in I} \\mathbf{A}_i$ onto $\\mathbf{A}_j$.\nA *subdirect product* of the family is a [[subalgebra|7CHG6SD365]] $\\mathbf{B}$ of $\\prod_{i \\in I} \\mathbf{A}_i$ such that $\\pi_j$ restricted to $B$ is [[surjective|01RVCMSN5N]] onto $A_j$ for every $j \\in I$.\nAn embedding $\\alpha \\colon \\mathbf{A} \\to \\prod_{i \\in I} \\mathbf{A}_i$ is *subdirect* when $\\alpha(\\mathbf{A})$ is a subdirect product.\n\n## Remarks\n\nLet $(\\theta_i)_{i \\in I}$ be a family of congruences on an algebra $\\mathbf{A}$.\nThe map $a \\mapsto (a/\\theta_i)_{i \\in I}$ is a subdirect embedding of $\\mathbf{A}$ into $\\prod_{i \\in I} \\mathbf{A}/\\theta_i$ if and only if $\\bigcap_{i \\in I} \\theta_i = \\Delta_A$.\n\nExample: the map $n \\mapsto (n \\bmod p)_p$ is a subdirect embedding of the ring $\\Z$ into $\\prod_{p \\text{ prime}} \\Z/p\\Z$.\nIt is a homomorphism, it is injective because a nonzero integer has only finitely many prime divisors, and each coordinate map $\\Z \\to \\Z/p\\Z$ is onto.\n\nNon-example: the subgroup $\\set{(x, 0) : x \\in \\Z/2\\Z}$ of $\\Z/2\\Z \\times \\Z/2\\Z$ is not a subdirect product.\nIts projection onto the second factor is $\\set{0}$, not all of $\\Z/2\\Z$.\n",
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