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  "body": "\n## Summary\n\nA factor congruence of an [[algebra|QJGPT2W0FT]] is a [[congruence|DCSXXT4SER]] that comes from splitting the algebra as a [[direct product|WJ73XYBNE4]] of two factors.\nIt is the kernel of the projection onto one factor.\nFactor congruences describe all the ways an algebra decomposes into a product of two algebras.\nAn algebra is [[directly indecomposable|6B1PSZ42MC]] when its only factor congruences are the trivial ones.\n\n## Explanations\n\n### L1 human\n\nIf an algebra is a product $\\mathbf{B} \\times \\mathbf{C}$, two pairs $(b, c)$ and $(b', c')$ can be compared by their first coordinates only.\nThe relation \"same first coordinate\" is a congruence, and it is a factor congruence.\nFactor congruences let one detect such a product splitting from inside the algebra, without knowing $\\mathbf{B}$ and $\\mathbf{C}$ in advance.\nFor example, $\\Z/6\\Z$ splits as $\\Z/2\\Z \\times \\Z/3\\Z$, and congruence modulo $2$ is the matching factor congruence.\n",
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