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  "body": "\n## Statement\n\nLet $\\mathbf{A}$ and $\\mathbf{B}$ be [[algebras|QJGPT2W0FT]] of the same signature and let $\\alpha \\colon \\mathbf{A} \\to \\mathbf{B}$ be a [[homomorphism|KBXSZC6BV1]].\nLet $\\ker \\alpha = \\set{(a, b) \\in A \\times A : \\alpha(a) = \\alpha(b)}$.\nLet $\\theta$ be a [[congruence|DCSXXT4SER]] on $\\mathbf{A}$ with $\\theta \\subseteq \\ker \\alpha$, and let $\\nu \\colon \\mathbf{A} \\to \\mathbf{A}/\\theta$ be the natural map onto the [[quotient algebra|N5KE7FH6CX]].\n\n1. The relation $\\ker \\alpha$ is a congruence on $\\mathbf{A}$.\n2. There is a unique homomorphism $\\beta \\colon \\mathbf{A}/\\theta \\to \\mathbf{B}$ with $\\beta \\circ \\nu = \\alpha$.\n3. The map $\\beta$ is injective if and only if $\\theta = \\ker \\alpha$.\n4. In particular, $a/\\ker \\alpha \\mapsto \\alpha(a)$ is an isomorphism from $\\mathbf{A}/\\ker \\alpha$ onto the subalgebra $\\alpha(\\mathbf{A})$ of $\\mathbf{B}$.\n\n## Remarks\n\nProof idea.\nCompatibility of $\\ker \\alpha$ follows from $\\alpha(f(a_1, \\dots, a_n)) = f(\\alpha(a_1), \\dots, \\alpha(a_n))$.\nPut $\\beta(a/\\theta) = \\alpha(a)$; this is well defined because $\\theta \\subseteq \\ker \\alpha$, and it is a homomorphism because $\\alpha$ and $\\nu$ are.\nUniqueness holds since $\\nu$ is surjective, and $\\beta(a/\\theta) = \\beta(b/\\theta)$ means $(a, b) \\in \\ker \\alpha$.\n\nExample: the ring homomorphism $\\Z \\to \\Z/n\\Z$ has kernel congruence modulo $n$, so the quotient of $\\Z$ by this congruence is isomorphic to $\\Z/n\\Z$.\nFor groups, part 4 is the [[first isomorphism theorem|0CXXJNG5B6]].\n\n## Explanations\n\n### L1 human\n\nA homomorphism may send several elements to the same place.\nIf you glue together exactly the elements with the same image, what is left is a copy of the image.\nSo every homomorphic image of an algebra is, up to isomorphism, a quotient of it.\n\n### L2 human\n\nPart 2 is the universal property of the quotient: maps out of $\\mathbf{A}/\\theta$ are the maps out of $\\mathbf{A}$ that do not separate $\\theta$-related elements.\nPart 1, together with $\\ker \\nu = \\theta$, shows that congruences are exactly the kernels of homomorphisms.\nWith the [[correspondence theorem|JG7WB75BHC]] it yields the other isomorphism theorems.\nIt is basic in the proof of [[Birkhoff's HSP theorem|5083DHSK82]], since it identifies the class operator $H$ with taking quotients.\n",
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