Reference no: EM131083778
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/akZ, where ai|ai+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∑∞anxn = a0 + a1x + a2x2 + . . . .
Recall that addition and multiplication are defined in essentially the same way as for polynomials.
(n=0∑∞anxn) + (n=0∑∞bnxn) = n=0∑∞(an + bn)xn
(n=0∑∞anxn) × (n=0∑∞bnxn) = n=0∑∞( k=0∑∞akbn-k)xn
(a) Prove that R[[x]] is a commutative ring with identity.
(c) Prove that n=0∑∞anxn 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.
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