homework8

# homework8 - Math 113 Section 5 Fall 2010 Homework#8 Due...

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Math 113 Section 5 Homework #8 Fall 2010 Due: Monday, November 1 1. Let R and R Í be rings and let R × R Í be their product. Determine which of the following are ring homomorphisms: (a) R R × R Í , r Ô→ ( r, 0) (b) R R × R , r Ô→ ( r,r ) (c) R × R Í R , ( r 1 ,r 2 ) r 1 (d) R × R R , ( r 1 ,r 2 ) r 1 r 2 (e) R × R R , ( r 1 ,r 2 ) r 1 + r 2 2. Section 7.3, Exercise 10: Decide which of the following are ideals in Z [ x ] : (a) the set of all polynomials whose constant term is a multiple of 3 (b) the set of all polynomials whose coeﬃcient of x 2 is a multiple of 3 (c) the set of all polynomials whose constant term, coeﬃcient of x and coeﬃcient of x 2 are zero (d) Z [ x 2 ] (i.e., the polynomials in which only even powers of x appear) (e) the set of polynomials whose coeﬃcients sum to zero (f) the set of polynomials p ( x ) such that p Í (0) = 0 , where p Í ( x ) is the usual ﬁrst
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