Question

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

Answer 1

If conclusion does not hold, thereexists an ideal I such that I2 ⊆ I. Then R/I hasa nonzero ideal I/I2 of square zero. This means I2 is not asemiprime ideal, so assumption does not hold.