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  "body": "\n## Summary\n\nA von Neumann regular ring is a ring in which every element $a$ has a weak inverse $x$, with $a x a = a$.\nFields, division rings, matrix rings over division rings, products of fields and Boolean rings are examples.\nJohn von Neumann introduced these rings in his work on continuous geometry.\nIn the commutative case they are exactly the reduced rings of Krull dimension $0$, and also the rings over which every module is flat.\nThe name is unrelated to regular local rings.\n\n## Explanations\n\n### L1 human\n\nIn a field every nonzero element has an inverse.\nA von Neumann regular ring asks for less: each element $a$ has a partner $x$ that undoes $a$ \"on its image\", in the sense that $a x a = a$.\nIn the product $\\Q \\times \\Q$ of two fields, the element $(2, 0)$ has no inverse, but $x = (\\frac12, 0)$ satisfies $a x a = a$.\nIn the integers, $2 x 2 = 2$ would need $4x = 1$, which has no solution, so $\\Z$ is not von Neumann regular.\n",
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