(a) Assume R is commutative. Let p(x) = anx" + an-1x"-1 + · ·. + a1x + ao be an element of the polynomial ring R[x]. Prove that p(x) is a unit in R[x] if and only if ao is a unit and a1 , a2, . .. , ɑn, are nilpotent in R. (b) Prove that the element (u, v) in direct product R x S of two rings R and S is unit if and only if u is unit in R and v is unit in S.
(a) Assume R is commutative. Let p(x) = anx" + an-1x"-1 + · ·. + a1x + ao be an element of the polynomial ring R[x]. Prove that p(x) is a unit in R[x] if and only if ao is a unit and a1 , a2, . .. , ɑn, are nilpotent in R. (b) Prove that the element (u, v) in direct product R x S of two rings R and S is unit if and only if u is unit in R and v is unit in S.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![(a) Assume R is commutative. Let p(x) = a,x" + an-1x"-1 + · · .
+ a1x + ao be an element of the polynomial ring R[x]. Prove that
p(x) is a unit in R[x] if and only if ao is a unit and a1 ,ɑ2, . , ɑn, are
nilpotent in R.
(b) Prove that the element (u, v) in direct product R x S of two rings R
and S is unit if and only if u is unit in R and v is unit in S.
%3|
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Transcribed Image Text:(a) Assume R is commutative. Let p(x) = a,x" + an-1x"-1 + · · .
+ a1x + ao be an element of the polynomial ring R[x]. Prove that
p(x) is a unit in R[x] if and only if ao is a unit and a1 ,ɑ2, . , ɑn, are
nilpotent in R.
(b) Prove that the element (u, v) in direct product R x S of two rings R
and S is unit if and only if u is unit in R and v is unit in S.
%3|
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