MATH 701 - 001 - Fall 2022

MATH 701 - 001 - Fall 2022

Assignment 8

Due Thursday, November 10, 2022

Problem 1

c.f. Exercise 7.1.7

The center of a ring \(R\) is \(\{z \in R \mid \forall r \in R, zr=rz\}\). Prove that the center of a ring is a subring, which contains the identity if it exists. Prove that the center of a division ring is a field.

Problem 2

c.f. Exercise 7.1.14

Let \(x\) be a nilpotent element of a commutative ring \(R\).

  1. Prove that \(x\) is either zero of a zero divisor.
  2. Prove that \(rx\) is nilpotent for all \(r \in R\).
  3. Prove that \(1+x\) is unit in \(R\).
  4. Deduce that the sum of a nilpotent element and a unit is a unit.

Problem 3

c.f. Exercise 7.2.3(c)

Let \(R[[x]]\) be the ring of formal power series in \(x\) with coefficients in \(R\). Prove that \(\sum_{n=0}^\infty a_nx^n\) is a unit in \(R[[x]]\) if and only if \(a_0\) is a unit in \(R\).

Solution

If \(A=\sum_{n=0}^\infty a_nx^n\) is a unit, then there exists \(B=\sum_{n=0}^\infty b_nx^n\) in \(R[[x]]\) such that \(AB=1\); thus \(a_0b_0=1\) in \(R\) and we conclude \(a_0\) is a unit.

Now suppose \(a_0\) is a unit. We define a sequence \(\{b_i\}\) in \(R\) recursively where \(b_0=a_0^{-1}\) and \[ b_{k} = -a_0^{-1} \sum_{i=1}^{k-1} a_i b_{k-i}\] for all integers \(k \ge 1\). We compute that \[\left(\sum_{n=0}^\infty a_nx^n \right)\left(\sum_{n=0}^\infty b_nx^n \right)=1,\] so \(A\) is a unit as desired.

Problem 4

c.f. Exercise 7.2.13

Let \(G\) be a finite group, let \(R\) be a commutative ring, and let \(RG\) be the corresponding group ring.

  1. If \(\mathcal{K}\) is a conjugacy class of \(G\), prove that \(\sum_{k \in \mathcal{K}} k\) is in the center of \(RG\).
  2. Let \(\mathcal{K}_1, \ldots, \mathcal{K}_r\) be the conjugacy classes of \(G\) and let \(K_i = \sum_{k \in \mathcal{K}_i} k\) be a corresponding element of \(RG\) for each \(\mathcal{K}_i\). Prove that \(\alpha \in RG\) is in the center of \(RG\) if and only if \(\alpha=\sum_{i=1}^r a_iK_i\) for some \(a_1,\ldots,a_r \in R\).

Solution

  1. Let \(K=\sum_{k \in \mathcal{K}} k\). For any \(g \in G\), the automorphism \(\phi_g(h)=hgh^{-1}\) of \(G\) restricts to a permutation of \(\mathcal{K}\). Thus \[g\left(\sum_{k \in \mathcal{K}} k \right)g^{-1}=\sum_{k \in \mathcal{K}} gkg^{-1} = \sum_{k \in \mathcal{K}} k\] and we conclude that \(gK=Kg\) for any \(g \in G\). An arbitrary element of \(RG\) has the form \(X=\sum_{g \in G} a_g g\) for \(a_g \in R\). We compute \[XK=\sum_{g \in G} a_g gK=\sum_{g \in G} a_g Kg=KX\] and conclude \(K\) is in the center \(Z(RG)\).

  2. If \(e\) is the identity of \(G\), then \(ae \in Z(RG)\) for all \(a \in R\). Since \(Z(RG)\) is a subring that contains every \(ae\) and every \(K_i\), we conclude that \(\sum_{i=1}^r a_iK_i \in Z(RG)\) for all \(a_1,\ldots,a_r \in R\). Conversely, suppose \(\alpha \in RG\) is not of the desired form, Then there must exist \(i\) such that \(g,k \in \mathcal{K}_i\) but \(\alpha_g \ne \alpha_k\) for the corresponding coefficients of \(\alpha\). There exists \(h \in G\) such that \(h g h^{-1} = k\). Consider \(\beta = h(\alpha)h^{-1}\) in \(RG\). We see that \(\beta_k = \alpha_g \ne \alpha_k\), so \(\alpha \ne \beta\) Thus \(h \alpha\ne \alpha h\) and we conclude \(\alpha\) is not in the center.

Problem 5

c.f. Exercise 7.3.23

Let \(S\) be a subring of \(R\) and let \(I\) be an ideal of \(R\). Prove that if \(S \cap I=0\) then \(S/I\) is a subring of \(R/I\) isomorphic to \(S\).

Problem 6

c.f. Exercise 7.3.35

Let \(I,J,K\) be ideals of a ring \(R\).

  1. Prove that \(I(J+K)=IJ+IK\) and \((I+J)K=IK+JK\).
  2. Prove that if \(J \subseteq I\) then \(I \cap (J+K) = J+(I \cap K)\).

Problem 7

c.f. Exercise 7.4.10

Assume \(R\) is commutative. Prove that if \(P\) is a prime ideal of \(R\) and \(P\) contains no zero divisors then \(R\) is an integral domain.

Problem 8

c.f. Exercise 7.4.27

Assume \(R\) be a finite commutative ring with identity. Prove that every prime ideal of \(R\) is a maximal ideal.