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Exercises 7.4 Exercises

1.

Find all of the ideals in each of the following rings. Which of these ideals are maximal and which are prime?
  1. \(\displaystyle {\mathbb Z}_{18}\)
  2. \(\displaystyle {\mathbb Z}_{25}\)
  3. \({\mathbb M}_2( {\mathbb R} )\text{,}\) the \(2 \times 2\) matrices with entries in \({\mathbb R}\)
  4. \({\mathbb M}_2( {\mathbb Z} )\text{,}\) the \(2 \times 2\) matrices with entries in \({\mathbb Z}\)
  5. \(\displaystyle {\mathbb Q}\)

2.

For each of the following rings \(R\) with ideal \(I\text{,}\) give an addition table and a multiplication table for \(R/I\text{.}\)
  1. \(R = {\mathbb Z}\) and \(I = 6 {\mathbb Z}\)
  2. \(R = {\mathbb Z}_{12}\) and \(I = \{ 0, 3, 6, 9 \}\)

3.

If \(R\) is a field, show that the only two ideals of \(R\) are \(\{ 0 \}\) and \(R\) itself.

4.

Prove that the associative law for multiplication and the distributive laws hold in \(R/I\text{.}\)

5.

Prove the Second Isomorphism Theorem for rings: Let \(I\) be a subring of a ring \(R\) and \(J\) an ideal in \(R\text{.}\) Then \(I \cap J\) is an ideal in \(I\) and
\begin{equation*} I / I \cap J \cong I + J /J. \end{equation*}

6.

Prove the Third Isomorphism Theorem for rings: Let \(R\) be a ring and \(I\) and \(J\) be ideals of \(R\text{,}\) where \(J \subset I\text{.}\) Then
\begin{equation*} R/I \cong \frac{R/J}{I/J}. \end{equation*}

7.

Prove the Correspondence Theorem: Let \(I\) be an ideal of a ring \(R\text{.}\) Then \(S \rightarrow S/I\) is a one-to-one correspondence between the set of subrings \(S\) containing \(I\) and the set of subrings of \(R/I\text{.}\) Furthermore, the ideals of \(R\) correspond to ideals of \(R/I\text{.}\)

8.

Let \(\{ I_{\alpha} \}_{\alpha \in A}\) be a collection of ideals in a ring \(R\text{.}\) Prove that \(\bigcap_{\alpha \in A} I_{\alpha}\) is also an ideal in \(R\text{.}\) Give an example to show that if \(I_1\) and \(I_2\) are ideals in \(R\text{,}\) then \(I_1 \cup I_2\) may not be an ideal.

9.

Let \(R\) be an integral domain. Show that if the only ideals in \(R\) are \(\{ 0 \}\) and \(R\) itself, \(R\) must be a field.

10.

Let \(R\) be a commutative ring. An element \(a\) in \(R\) is nilpotent if \(a^n = 0\) for some positive integer \(n\text{.}\) Show that the set of all nilpotent elements forms an ideal in \(R\text{.}\)

11.

Let \(R\) be a ring and let \(u\) be a unit in \(R\text{.}\) Define a map \(i_u \colon R \rightarrow R\) by \(r \mapsto uru^{-1}\text{.}\) Prove that \(i_u\) is a homomorphism from \(R\) to itself. A homomorphism from a ring \(R\) to itself is called an automorphism of \(R\) and an automorphism like \(i_u\) is called an inner automorphism.

12. The Chinese Remainder Theorem for Rings.

Let \(R\) be a ring and \(I\) and \(J\) be ideals in \(R\) such that \(I+J = R\text{.}\)
  1. Show that for any \(r\) and \(s\) in \(R\text{,}\) the system of equations
    \begin{align*} x & \equiv r \pmod{I}\\ x & \equiv s \pmod{J} \end{align*}
    has a solution.
  2. In addition, prove that any two solutions of the system are congruent modulo \(I \cap J\text{.}\)
  3. Let \(I\) and \(J\) be ideals in a ring \(R\) such that \(I + J = R\text{.}\) Show that there exists a ring isomorphism
    \begin{equation*} R/(I \cap J) \cong R/I \times R/J. \end{equation*}

13.

An ideal of a commutative ring \(R\) is said to be finitely generated if there exist elements \(a_1, \ldots, a_n\) in \(R\) such that every element \(r \in R\) can be written as \(a_1 r_1 + \cdots + a_n r_n\) for some \(r_1, \ldots, r_n\) in \(R\text{.}\) Prove that \(R\) satisfies the ascending chain condition if and only if every ideal of \(R\) is finitely generated.

14.

Let \(D\) be an integral domain with a descending chain of ideals \(I_1 \supset I_2 \supset I_3 \supset \cdots\text{.}\) Suppose that there exists an \(N\) such that \(I_k = I_N\) for all \(k \geq N\text{.}\) A ring satisfying this condition is said to satisfy the descending chain condition, or DCC. Rings satisfying the DCC are called Artinian rings, after Emil Artin. Show that if \(D\) satisfies the descending chain condition, it must satisfy the ascending chain condition.
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