[代數] 幾題代數

[hotplushot (熱加熱)]

1.find a finite ring with characteristic 4.

2.find a infinite ring with characteristic 4.

3.
f:R→S is a homo of rings with identity
and u is a unit in R such that f(u) is a unit in S,

then f(1_R)=1_S and f(u^-1)=f(u)^-1

4.
|G|=(p^n)q, with p>q(p, q are primes), then G contians a unique normal

subgroup of index q.

5.找一個環(包含identity u) 其子環的identiy不為u.
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◆ From: 111.252.193.181
→ lockheart   :3.f(1_R)=f((1_R)^2)=f(1_R)^2. Since 1_R is a unit
→ lockheart   :in R, f(1_R) is a unit in S. By cancellation law
→ lockheart   :we can find that f(1_R)=1_S. Moreover 1_S=f(1_R)
→ lockheart   :=f(u*u^-1)=f(u)*f(u^-1) for some unit u in R,
→ lockheart   :f(u^-1)=f(u)^-1.
→ lockheart   :4. R_1*R_2 and {0}*R_2
感謝!!!
※ 編輯: hotplushot      來自: 111.252.193.181      (06/23 19:53)
→ Vulpix      :4. Sylow's Thm.
→ Vulpix      :1. Z/(4)    2. 無限多個Z/(4)直積
→ RoamingGhost:or any polynomial ring over Z/4Z, like Z/4Z [t]