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  "body": "\n## Summary\n\nA simple algebra is a nontrivial [[algebra|QJGPT2W0FT]] whose only [[congruences|DCSXXT4SER]] are the two trivial ones.\nSo it has no proper nontrivial [[quotient algebras|N5KE7FH6CX]]: every homomorphism out of it is injective or constant.\nThe notion covers [[simple groups|0YMYMNQ0QZ]] and simple rings, and it is the basic building block in the study of varieties.\n\n## Explanations\n\n### L1 human\n\nA simple algebra cannot be shrunk by gluing elements together, except by gluing everything into one point.\nFor a group, gluing is controlled by normal subgroups, so a simple group is a group with no normal subgroups other than the trivial one and itself.\nFor example, $\\Z/5\\Z$ is a simple group, while $\\Z/6\\Z$ is not, since it can be glued down to $\\Z/2\\Z$.\nEvery field is simple as a ring, because its only ideals are $\\set{0}$ and the whole field.\n",
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