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  "body": "\n## Summary\n\nA semilocal ring is a ring $R$ whose quotient $R/J(R)$ by the [[Jacobson radical|XA0DCNH7WY]] is a [[semisimple ring|NY25HXA478]].\nFor a [[commutative ring|0310REJJY8]] this is the same as having finitely many maximal ideals, the [[commutative notion|6DYAGW68AV]].\nLeft Artinian rings and [[local rings|MZ02BKPHJA]] are semilocal.\nSemilocal rings are the starting point for [[semiperfect|ZRC2S3ECFK]], [[perfect|49Y9Q949PK]] and [[semiprimary|XVP517QRH5]] rings, which add conditions on $J(R)$.\n\n## Explanations\n\n### L1 human\n\nDividing a ring by its Jacobson radical removes the part that simple modules cannot see.\nFor a semilocal ring, what remains is as simple as possible: a finite product of matrix rings over division rings.\nFor example, the ring of upper triangular $2 \\times 2$ matrices over a field is semilocal, since modulo its radical it is a product of two copies of the field.\nThe integers are not semilocal, since they have infinitely many maximal ideals.\n",
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