# ةثلاثلا ةلحرملا/تاقلحلا ةدامل لولأا لصفلا ناحتما ةلئسأ جذومن

```‫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&times;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&times;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 &gt;0. Show that in the
additive group (R, +) all nonzero elements have the same order n.
6. State and prove the &quot; Fundamental Homomorphism Theorem&quot; .
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
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