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. --

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