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المرحلة الثالثة/نموذج أسئلة امتحان الفصل األول لمادة الحلقات Part A: Chose the right answer. 1. The following statement is true in any ring: a) a nonzero idempotent element is nilpotent. b) a nonzero idempotent cannot be nilpotent. c) every nonzero nilpotent element is idempotent. d) every nilpotent element is idempotent. 2. In the ring Z 10 , the following statement is not true: a) [6] is a zero divisor. b) [3] is an invertible element. c) there is no nonzero nilpotent element. d) there is exactly three idempotent elements. 3. The following mapping is not a ring homomorphism: a) f : Z→ Z e , f(a) = 2a. b) f : Z×Z→ Z, f (a, b) = a. c) f : R→ R, f (a + b 2 ) = a - b 2 where R= { a + b 2 | a, b Z}. a 0 d) f : Z×Z→ M 2 (Z), f (a, b) = . 0 b 4. Let f : R→S be a ring homomorphism, where both R and S are rigs with identity. Then the following statement is not true: a) f(1 R ) =1 S . b) f(A) is a subring of S, if A is a subring of R. c) f 1 (B) is a subring of R, if B is a subring of S. d) f 1 (J) is an ideal of R if J is an ideal of S. Part B: 5. Let R be an integral domain, and char R =n >0. Show that in the additive group (R, +) all nonzero elements have the same order n. 6. State and prove the " Fundamental Homomorphism Theorem" . 7. Prove: any finite integral domain is a field. 8. Let R be a commutative ring with identity and I a maximal ideal of R. Show that R/I is a field. Good Luck