# Answer to Question #23255 in Abstract Algebra for Hym@n B@ss

Question #23255

Show that in a right artinian ring R, every prime ideal p is maximal.

Expert's answer

*R/*p is semiprime and right artinian, so it issemisimple. Since

*R/*p is in fact prime, it can have only one simplecomponent. Therefore,

*R/*p is simple, so p is a maximal ideal.

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