Unformatted text preview: First read the deﬁnition of the product of two ideals given on page 247 in the text, since we did not cover this in class yet. We say that an ideal I is nilpotent if I n = 0 for some n ≥ 1. (Note that this is stronger than saying that x n = 0 for all x ∈ I ; it means b 1 b 2 ...b n = 0 for all b i ∈ I .) 1 . (a). Prove that if N = ( a 1 ,...,a m ) = a 1 R + ··· + a m R is the ideal generated by ﬁnitely many elements a i ∈ R , and each a i is nilpotent in R , then N is a nilpotent ideal. (b). Prove that a polynomial p ( x ) ∈ R [ x ], say p ( x ) = a + a 1 x + ··· + a n x n , is nilpotent in R [ x ] if and only if each a i is nilpotent in R for all 0 ≤ i ≤ n . (c). Prove that a polynomial p ( x ) ∈ R [ x ], say p ( x ) = a + a 1 x + ··· + a n x n , is a unit in R [ x ] if an only if a is a unit in R , and a 1 ,...,a n are nilpotent in R . 1...
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This note was uploaded on 12/11/2011 for the course MATH 200a taught by Professor Bunch,j during the Fall '08 term at UCSD.
 Fall '08
 Bunch,J
 Math

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