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  "body": "\n## Summary\n\nA simple ring is a nonzero ring whose only [[two-sided ideals|0CD87SFM2N]] are $0$ and the whole ring.\nIt cannot be broken into smaller rings by passing to quotients.\nFields, [[division rings|0EA5BTA0CX]] and full [[matrix rings|0FHVPRAE60]] over them are simple.\nBy Wedderburn's theorem, the simple Artinian rings are exactly the matrix rings over division rings.\n\n## Explanations\n\n### L1 human\n\nFor a commutative ring, having no ideals except $0$ and itself means being a field.\nFor noncommutative rings, one asks this only of two-sided ideals, and many more rings qualify.\nThe ring of $2 \\times 2$ real matrices is simple, although it has zero divisors and one-sided ideals.\nSimple rings play the role among rings that simple groups play among groups.\n",
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