# ahw9 - Math 4124 Wednesday, April 27 Ninth Homework...

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Math 4124 Wednesday, April 27 Ninth Homework Solutions 1. Exercise 7.3.10. Decide which of the following are ideals of the ring Z [ x ] . Let I be the relevant subset of Z [ x ] . (a) The set of all polynomials whose constant term is a multiple of 3. Yes. Obviously 0 I and I is an abelian group under addition. Finally suppose f = a 0 + a 1 x + ··· + a m x m I and g = b 0 + b 1 x + ··· + b n x n Z [ x ] . Then 3 divides a 0 and fg = a 0 b 0 + ( a 1 b 0 + a 0 b 1 ) x + ··· + a m b n x m + n . Since 3 divides a 0 b 0 , we see that fg I and we have proven that I is an ideal. (b) The set of all polynomials whose coefﬁcient of x 2 is a multiple of 3. No. x I yet x 2 / I , so I is not even a subring. (c) The set of all polynomials whose constant term, coefﬁcient of x and coefﬁcient of x 2 are zero. Yes. This is just the ideal x 3 Z [ x ] . (d) Z [ x 2 ] . No. 1 I but 1 x / I (on the other hand, I is a subring).

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## ahw9 - Math 4124 Wednesday, April 27 Ninth Homework...

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