Section 4 Problem Set 4

Due: Wednesday April 28, midnight CST. Time will be provided in class to work on these and discuss them with your peers. This problem set has 4 problems.

4.1 Boolean Rings

Let \(R\) be a ring in which \(a^2 = a\) for every \(a \in R.\)

  1. Prove that \(a = -a\) in this ring.

  2. Prove that \(R\) is commutative. (Hint: cleverly use the \(a^2 = a\) property and don’t forget that we have addition in \(R\) but we don’t have cancellation).

  3. Prove that if \(R\) has unity and \(a \not= 0,1\) then \(a\) is a zero divisor.

  4. \(\mathbb{Z}_2 =\{0,1\}\) is a Boolean ring with 2 elements. Give an example of a Boolean ring with unity having 4 elements.

4.2 Subsets, Subrings, and Ideals

Each example below describes a subset \(S \subseteq R\). Determine (with proof) whether \(S\) is “just” a subset or is a subring or is an ideal.

  1. \(R = \mathbb{Z}[x]\) and \(S\) is the set of polynomials with 0 constant term: \(S = \{ p(x) = a_0 + a_1 x + \cdots + a_n x^n \in \mathbb{Z}[x] \mid a_0 = 0\}\).

  2. \(R = \mathbb{Z}[x]\) and \(S\) is the set of polynomials with an even constant term: \(S = \{ p(x) = a_0 + a_1 x + \cdots + a_n x^n \in \mathbb{Z}[x] \mid a_0 \in 2 \mathbb{Z}\}\).

  3. \(R = \mathbb{Z}[x]\) and \(S\) is the set of polynomials with a zero quadratic term: \(S = \{ p(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots + a_n x^n \in \mathbb{Z}[x]\mid a_2 = 0 \}\).

  4. \(R\) is a commutative ring, \(a \in R\), and \(S = \{ r \in R \mid r a = 0 \}\).

  5. \(R\) is a ring and \(S = \{ s \in R \mid s r = r s \text{ for all } r \in R\}\).

Hint: you might want to use some ring properties in Definitions.

4.3 Gaussian Quotient Ring

Let \(I = \langle 2 + 2 i \rangle \subseteq \mathbb{Z}[i]\). That is, \(I\) is a principal ideal in the ring of Gaussian integers.

  1. List the distinct cosets in \(\mathbb{Z}[i]/I\).

  2. Find the units in \(\mathbb{Z}[i]/I\). Give each unit with its inverse.

  3. If \(\mathbb{Z}[i]/I\) has zero divisors, give an example of one.

  4. Where in the Venn diagram of rings does \(\mathbb{Z}[i]/I\) live?

  5. Is \(I\) a prime ideal? Is \(I\) a maximal ideal? In each case, say why or why not.

4.4 Ideal Radicals

Let \(R\) be a commutative ring, let \(A\) be any ideal of \(R\), define the radical of \(A\) to be \[ N(A) = \{ r \in R \mid r^n \in A \text{ for some positive integer } n\} \] (the integer \(n\) can be different for different choices of \(r\)).

  1. Prove that \(N(A)\) is an ideal of \(R\).1
  2. Give an example where \(N(A) = A\) and an example where \(A \subsetneqq N(A)\).
  3. The nilradical of \(R\) is \(N(\{0\})\). Find (with proof) the nilradical of \(R/N(A)\) for any ideal \(A\). That is, state what it is and then prove your statement. Here is a movie ( mov|pdf) that might clarify things.

  1. For one of the steps consider using the binomial theorem: \((a+b)^n = \displaystyle{\sum_{k=0}^n \binom{n}{k} a^k b^{n-k}}\).↩︎