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  "body": "\n## Definition\n\nLet $\\mathbf{A}$ be an [[algebra|QJGPT2W0FT]] whose universe has at least two elements.\nThe algebra $\\mathbf{A}$ is *directly indecomposable* when it is not isomorphic to any [[direct product|WJ73XYBNE4]] $\\mathbf{B} \\times \\mathbf{C}$.\nHere $\\mathbf{B}$ and $\\mathbf{C}$ range over algebras of the same signature with at least two elements each.\n\n## Remarks\n\nEquivalently, the only [[factor congruences|6N268PHKQA]] of $\\mathbf{A}$ are $\\Delta_A$ and $\\nabla_A$.\nEvery finite algebra is isomorphic to a direct product of finitely many directly indecomposable algebras, by induction on the number of elements.\n\nExample: an algebra $\\mathbf{A}$ with a prime number $p$ of elements is directly indecomposable.\nIf $\\mathbf{A} \\cong \\mathbf{B} \\times \\mathbf{C}$, then $p = \\abs{B} \\cdot \\abs{C}$, so $\\abs{B} = 1$ or $\\abs{C} = 1$.\n\nNon-example: the group $\\Z/6\\Z$ is not directly indecomposable.\nThe map $k \\mapsto (k \\bmod 2, k \\bmod 3)$ is an isomorphism onto $\\Z/2\\Z \\times \\Z/3\\Z$, and both factors have at least two elements.\n\nEvery subdirectly irreducible algebra is directly indecomposable, but not conversely: the ring $\\Z$ is directly indecomposable and not subdirectly irreducible.\n",
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