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

Modern Algebra Ring Theory (XXXIX)
Polynomial Rings over Commutative Rings
Unit Element

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

Solution Summary

This solution is comprised of a detailed explanation of Polynomial Rings over Commutative Rings. It contains step-by-step explanation that R[x] is a commutative ring with unit element whenever R is.

... over Commutative rings Integral Domain Unit Element Unit of a Commutative Ring. If R is an integral domain with unit element, prove that any unit in R[x] is a ...

...rings Integral Domain Unit Element Unit of a Commutative Ring. If R is an integral domain with unit element, prove that any unit in R[x] must already be a unit...

... Proof: (⟹) Suppose R is a nonzero local ring. ... (⟸) Suppose R is a commutative ring with 1≠0 such that for all x∈R, either x or 1-x is a unit. ...

... Or, Prove that if R is an integral domain, then R[x1 ... said to be a polynomial ring over a commutative ring R and it is usually denoted by the symbol R[x ]. ...

...Rings Unit Element Integral Domain Degree of a Polynomial Zero-divisor of a commutative ring. ... If R is an integral domain, then so is R[x ] . Or, Prove that if ...

...prove that there is an element b ≠ 0 in R... divisor of a commutative ring In a commutative ring R , a ≠ 0 ... divisor if there exists b ≠ 0 ∈ R such that ab ...

... For any commutative ring R, it can be proved that a polynomial f(x) = a(sub 0) + a(sub 1)x +...+a(sub n)x^n element R[x] is a unit in R[x] if and only if a(sub ...

...prove that ( f (x... For if F [x ] is a Euclidean ring ( or, if F [x ] is a commutative ring with unit... Thus r ( x ) = ao + a1 x + a 2 x + ... + an −1 x 2 n −1 ...

...Prove that ) is a (x 2 +1 field isomorphic to the ... if F [x ] is a Euclidean ring (or a commutative ring with unit... f ( x ) = t ( x ) x 2 + 1 + r ( x ) , where r...