课程名称︰代数导论一
课程性质︰必修
课程教师︰庄武谚
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2014/01/06
考试时限（分钟）：Take home quiz
试题 :
代数导论第七次小考
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.