Your Name Math 171 Homework 8 Due: Nov. 12, 2014 1 Prove that any finite abelian group is isomorphic to a product of cyclic groups Z/a1 Z × Z/a2 Z × · · · × Z/ak Z where ai | ai+1 for all i. 1 2 A ring R is called Boolean if a2 = a for all a ∈ R. Prove that every Boolean ring is commutative. 2 3 Let R be a commutative ring. We say x ∈ R is nilpotent if there is a positive integer n ≥ 1 for which x n = 0. Let x be a nilpotent element of a ring R. (a) Prove that x is either zero or is a zero divisor. (b) Prove that rx is nilpotent for all r ∈ R. (c) Prove that 1 + x is a unit in R. (d) Deduce that the sum of a nilpotent element and a unit is a unit. 3 4 Let R be a commutative ring with identity and define the set R[[ x ]] of formal power series in x with coefficients from R to be all formal infinite sums ∞ ∑ a n x n = a0 + a1 x + a2 x 2 + . . . . n =0 Recall that addition and multiplication are defined in essentially the same way as for polynomials. ! ! ∞ ∑ an x n ∞ n =0 ∞ ∑ an x n n =0 × = n =0 ∞ ! ∞ ∑ bn x n + ∑ bn x n n =0 ! n =0 ∑ ( a n + bn ) x n = ∞ n n =0 k =0 ∑ ∑ a k bn − k ! xn (a) Prove that R[[ x ]] is a commutative ring with identity. n (c) Prove that ∑∞ n=0 an x is a unit in R [[ x ]] if and only if a0 is a unit in R. (d) Prove that if R is an integral domain then R[[ x ]] is an integral domain. 4 5 Consider the following elements of the integral group ring ZS3 : α = 3(1 2) − 5(2 3) + 14(1 2 3) and β = 6(1) + 2(2 3) − 7(1 3 2) (where (1) is the identity of S3 ). Compute the following elements: (a) α + β, (b) 2α − 3β, (c) αβ, (d) βα, (e) α2 . 5 6 Let n ≥ 2 be an integer. Prove every non-zero element of Z/nZ is a unit or a zero divisor. 6