If R is an integral domain with unit element, prove that any unit in R[x] is a unit in R and any unit in R is also a unit in R[x].

Related BrainMass Solutions

• If R is an integral domain with unit element, prove that any unit in R[x] must already be a unit in R.

  127644 If R is an integral domain with unit element, prove that any unit in R[x] must already be a unit in R.

• If R is an Integral Domain, then so is R[x].

  138887 If R is an Integral Domain, then so is R[x]. If R is an integral domain, then so is R[x]. Prove that if R is an integral domain, then R[x] is also an integral domain. See attached file for full problem description.

• Finite ring proofs

  Suppose that R has identity, prove that if x E R is not a zero divisor, then it is a unit Proof: (a) Since R is finite, assume |R| = n, then for any non-zero element x in R, we have x^n = x.

• Integral Domains and Fields : Embedding Theorem

  Proof: R is an integral domain, it means R is a commutative ring with the multiplicative unit e. Suppose R C F, where F is a field. Now we construct a map f: Q->F. For any element x=a/b in Q, where a,b are integers, we define f(x)=f(a/b)=ae/be.

• Ring proofs

  Prove that the polynomial f(x) is a unit in R[x] if and only if f(x) is a nonzero constant which is a unit in R. b.

• Ring and fields

  To see this another way, recall that if R is an integral domain, and S = R - {0}, then Q : = R_S is the field of fractions of R; and if F is a field containing an isomorphic copy R' of R, then the subfield of F generated by R' is isomorphic to

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  127643 Prove that if R is a commutative ring with unit element then R[x] is also a commutative ring with unit element.

• If R is an integral domain, then so is R[x_1, x_2, ... , x_n] . Or, Prove that if R is an integral domain, then R[x_1, x_2, ... , x_n] is also an integral domain.

  Or, Prove that if R is an integral domain, then R[x_1, x_2, ... , x_n] is also an integral domain.

• Nilpotents and Zero Divisors

  in R d)Deduce that the sum of a nilpotent element and a unit is a unit Proof: a) If x is not 0, since x^m=0, then x*(x^(m-1))=0. let y=x^(m-1), then xy=0.