Commutative ring with no non-zero nilpotent elements
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Let R be a commutative ring with no non-zero nilpotent elements (that is, a^n = 0 implies a = 0).
If f(x) = a_0 + a_1x + a_2x^2 +...+ a_mx^m in R[x] is a zero-divisor, prove that there is an element b is not equal to 0 in R such that ba_0 = ba_1 = ba_2 = ...=ba_m = 0.
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Solution Summary
This solution is comprised of a detailed explanation of Zero divisor of Commutative Rings.
It contains step-by-step explanation of the problem that if f(x) = a_0 + a_1x + a_2x^2 +...+ a_mx^m in R[x] is a zero-divisor,
then there is an element b is not equal to 0 in R such that ba_0 = ba_1 = ba_2 = ...=ba_m = 0 where R is a commutative ring with no non-zero nilpotent elements (that is, a^n = 0 implies a = 0). Notes are also given at the end.
Solution Preview
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Modern Algebra
Ring Theory (XLIII)
Polynomial Rings over Commutative rings
Zero-divisor of a Commutative ring
Nilpotent Elements
By:- Thokchom Sarojkumar Sinha
Let be a commutative ring with no non-zero nilpotent elements ( that is, implies ).
If in is a zero-divisor,
prove that there is an element in such that .
Solution:- Let be a zero-divisor .
Then there exists non-zero polynomial such that
where
Then ------------------------------------------------------------------------(1)
If then
For is a commutative ring.
by using (1).
-----------------------------------------------------------------(2)
and
For if since has no non-zero nilpotent elements.
Again,
For is commutative.
by using (1).
For by (2).
...
Education
- BSc, Manipur University
- MSc, Kanpur University
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