, r,~1oV<"
中国科学技术大学 a-`OE"
2016年秋季博士资格考试试卷 ]Tje6i F
"dKYJ&$
~N i#xa
代数学 >};6>)0
u#+p6%?k
1." role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">1.1.(40分) 考虑形式幂级数环 C[[x]]={a0+a1x+a2x2+⋯∣ai∈C}" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">C[[x]]={a0+a1x+a2x2+⋯∣ai∈C}C[[x]]={a0+a1x+a2x2+⋯∣ai∈C} 考虑 2" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">22 阶全矩阵环 R=M2(C[[x]])" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">R=M2(C[[x]])R=M2(C[[x]])． O1o.^i$-M
(1) 证明 C[[x]]" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">C[[x]]C[[x]] 为 Noether 整环； ,tEvz
(2) 描述 C[[x]]" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">C[[x]]C[[x]] 全部的有限生成不可分解模，并给出论证； 0I.9m[<Fc
(3) 给出环 R" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">RR 全部的双边理想，并给出论证； }T?
MWcG4
(4) 描述 R" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">RR 上全部的有限生成不可分解左模，以及这些模的自同态环． 2u[:3K-@,
<JA`e+Bi
2." role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">2.2.(40分) 将 Abel 群与 Z" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">ZZ-模等同起来，考虑 Abel 群 G=Z3⊕Z" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">G=Z3⊕ZG=Z3⊕Z．
c|N!ZYJI
(1) 列出群 G" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">GG 的全部子群，并给出论证； Dm@wTt8N(
(2) 将 G" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">GG 的每个商群都分解成不可分解群的直和，并给出论证； \ B~9Ue!
(3) 列出群 G" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">GG 的全部直和项，并给出论证； ZCQ7xQD
(4) 描述 G" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">GG 的自同构群． #]s>
回顾：Abel 群 G" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">GG 的子群 A" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">AA 称为直和项，若存在另一子群 B" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">BB 满足 G=A+B" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">G=A+BG=A+B 以及 A∩B={0}" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">A∩B={0}A∩B={0}． (D'Z 4Y
Sx,O )
3." role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">3.3.(20分) 具体给出代数同构 r9Ux=W\
CS3→∼C×C×M2(C)，" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; text-align: left; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">CS3−→～C×C×M2(C)，CS3→～C×C×M2(C)， 其中 CS3" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">CS3CS3 为 S3" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">S3S3 的群代数；并给出相应的论证． >'i
d/
提示：利用不可约复表示． v+Vp ak9|
|
