Question #24900

For a ring R, prove that: if every ideal of R is semiprime, then every ideal I of R is idempotent.

Expert's answer

If conclusion does not hold, thereexists an ideal *I *such that *I*2 *⊆* *I*. Then *R/I *hasa nonzero ideal *I/I*2 of square zero. This means *I*2 is not asemiprime ideal, so assumption does not hold.

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