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
...
Education
- BSc, Manipur University
- MSc, Kanpur University
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