課程名稱︰代數導論一
課程性質︰必修
課程教師︰莊武諺
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2014/01/06
考試時限（分鐘）：Take home quiz
試題 :
代數導論第七次小考
INTRODUCTION TO ALGEBRA I - TAKE HOME QUIZ VII
Please submit your answer sheet to me at Astro/Math 403 before 5pm,Jan 6
2014. If I am not in, slip the answer sheeet under my door. Perfect scores
will contribute 3 points to your final only when your scores are under 60.
The quiz starts from here.
(1) (12 points) Let I and J be ideals of ring R.
(i) Prove that I + J is the smalleest ideal of R containing both I and J
(ii) Prove that IJ is an ideal contained in I ∩ J.
(iii) Give an example where IJ ≠ I ∩ J.
(2) (6 points) Please classify nonabelian groups with 8 elements.
(3) (6 points) Let G be a group of order 224. Show that G is not simple.
(4) (6 points) Let R be a commutative ring with 1. Prove that the principal
ideal (x) generated by x in the polynomial ring R[x] is a prime ideal if
and only if R is an integral domain. Moreover (x) is a maximal ideal if
and only if R is a field.