Math 171: Abstract Algebra, Fall 2014- Assignment 8

1. Prove that any finite abelian group is isomorphic to a product of cyclic groups

Z/a₁Z × Z/a₂Z × · · · × Z/a_kZ, where a_i|a_i+1 for all i.

2. A ring R is called Boolean if a² = a for all a ∈ R. Prove that every Boolean ring is commutative.

3. Let R be a commutative ring. We say x ∈ R is nilpotent if there is a positive integer n ≥ 1 for which xⁿ = 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.

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

_n=0∑^∞a_nxⁿ = a₀ + a₁x + a₂x² + . . . .

Recall that addition and multiplication are defined in essentially the same way as for polynomials.

(_n=0∑^∞a_nxⁿ) + (_n=0∑^∞b_nxⁿ) = _n=0∑^∞(a_n + b_n)xⁿ

(_n=0∑^∞a_nxⁿ) × (_n=0∑^∞b_nxⁿ) = _n=0∑^∞( _k=0∑^∞a_kb_n-k)xⁿ

(a) Prove that R[[x]] is a commutative ring with identity.

(c) Prove that _n=0∑^∞a_nxⁿ 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.

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.

6. Let n ≥ 2 be an integer. Prove every non-zero element of Z/nZ is a unit or a zero divisor.