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**Unformatted text preview: **if a 2 = a . Prove that if R is commutative and has characteristic 2, then the idempotent elements form a subring. 5. If R is a nite commutative ring with unity, prove that every maximal ideal of R is a prime ideal of R . 6. Let R be a commutative ring with unity that has the property that a 2 = a for all a R . Let I be a prime ideal in R . Show that the order of R/I must be 2. 7. Show that Z 3 [ x ] / h x 2 +1 i is ring-isomorphic to Z 3 [ i ] = { a + bi | a,b Z 3 } . Is the same statement true if we replace Z 3 with Z 7 ? 1...

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