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

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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.

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Solution Summary

This solution is comprised of a detailed explanation of the properties of polynomial rings over commutative rings or integral domains.It contains step-by-step explanation and a reference.

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Modern Algebra
Ring Theory (XLIX)
Polynomial Rings over Commutative Rings
Unit Element
Integral Domain
Degree of a Polynomial
Zero-divisor of a commutative ring

By:- Thokchom Sarojkumar Sinha

If is an integral domain, then so is .
Or,
Prove that if is an integral domain, then is also an integral domain.

Solution:- Let be an integral domain.

To prove that is an integral domain.

Let

(1) is in .
For

where for each
that is,
Then .

(2) is commutative.
For

(3) is associative,
that is,
For

...

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