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  "body": "\n## Summary\n\nA directly indecomposable algebra is a nontrivial [[algebra|QJGPT2W0FT]] that is not isomorphic to a [[direct product|WJ73XYBNE4]] of two nontrivial algebras.\nEvery finite algebra is a direct product of finitely many directly indecomposable ones.\nThe notion is weaker than [[subdirect irreducibility|DASF939439]], which forbids also the more flexible subdirect decompositions.\n\n## Explanations\n\n### L1 human\n\nSome algebras split into two independent parts, like $\\Z/6\\Z$, which behaves as $\\Z/2\\Z$ and $\\Z/3\\Z$ side by side.\nA directly indecomposable algebra admits no such splitting.\nAny algebra with a prime number of elements is directly indecomposable, because a product of two nontrivial algebras has a composite number of elements.\nThe ring $\\Z$ is directly indecomposable too, even though it is not subdirectly irreducible.\n",
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