Prove that for a commutative ring the cancellation property of ... | Filo
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Let R be a commutative ring with unity. If I is a prime ideal of R, prove that I[x] is a prime ideal of R[x]. | Numerade
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SOLVED: Let R be a commutative ring with unity and let U(R) denote theset of units of R. Prove that U(R) is a group under themultiplication of R. (This group is called the group of units ofR.)
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SOLVED: Let R be a ring such that x 3 = x for all x ∈ R. Prove that R iscommutative by proving the followings: (In the proof, you can usethe facts in the previous steps)(a) Prove that for a, b ∈ R such that ab = 0, then ba = 0.(b) If c ∈ R is such that c 2 = c, prove that cx = xc holds forall x ∈ R.(c) Prove that for any x ∈ R, x 2y = yx2 for any y ∈ R.(d) If c 2 = 2c(= c c) for some c, then prove that cy = yc forany y ∈ R.(e) For any x ∈ R, (x x 2 )y = y(x x 2 ) for any y ∈R.(f) Prove tha
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