Re: [代數] ring & ideal
※ 引述《jayfrog (宅可宅 非常宅)》之銘言：
: 1.
: Let R be a ring such that x^2=x for all x belong R.
: Show that R is commutative.
(1) (a+1)=(a+1)^2=a^2+2a+1=a+2a+1 => 2a=0
(2) (a+b)=(a+b)^2=a^2+ab+ba+b^2=a+ab+ba+b => ab+ba=0
by(1) => 2(ab)=0 => ab=-ab
by(2) => ab+ba=-ab+ba=0 => ab=ba
: 2.
: Let R be a commutative ring. If Mis an ideal, abbreviate MM by M^2.
: Let M_1, M_2 be two ideals such that M_1+M_2=R.
: Show that M_1 ^2 + M_2 ^2 =R
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