Homework #6

1. In each of the following rings, \( R \): state the characteristic of the ring, give an example of a unit (other than \( 1\)), and give an example of a zero divisor. If no unit exists, explain why not. If no zero divisors exist, explain why not.
   
   
   1. \( R = \mathbb{Z}_4 \oplus \mathbb{Z}_6 \)
      
      
   2. \( R = 3\mathbb{Z} = \{ 3k \,|\, k \in \mathbb{Z} \} \) (multiples of 3)
      
      
   3. \( R = (\mathbb{Z}_3)^{2 \times 2} \) (\(2 \times 2\) matrices with entries in \( \mathbb{Z}_3 \))
      
      
2. Recall \( R \oplus S \) is the direct product of the rings \( R \) and \( S \).
   
   
   1. Suppose \( R \) and \( S \) have 1's. Then show \( R \oplus S \) is also a ring with 1.
      
      
   2. Suppose that \( R \) and \( S \) are integral domains. Show that \( R \oplus S \) is not an integral domain.
      
      
   3. Can \( R \oplus S \) be an integral domain? a field? Be careful: We are not assuming that \( R \) and \( S \) are integral domains or fields so the previous part does not apply directly.
      
      
   4. Let \( R \) and \( S \) be rings with 1. Prove \( U(R \oplus S) \cong U(R) \oplus U(S) \) (i.e. the group of units of \( R \oplus S \) is isomorphic to the direct product of the group of units of \( R \) and the group of units of \( S \)).
      Hint: You will need to write down a (group) isomorphism.
      
      
3. Let \( \displaystyle{ S = \left\{ \begin{bmatrix} x & 0 \\ x & 0 \end{bmatrix} \,\Big|\, x \in \mathbb{R} \right\} } \).
   
   
   1. Show \( S \) is a subring of \( \mathbb{R}^{2 \times 2} \) (the ring of \( 2 \times 2 \) real matrices).
      
      
   2. Is \( S \) commutative?
      
      
   3. Does \( S \) have a unity (i.e. a multiplicative identity)? If so, what is it?
      
      
   4. Is \( S \) an integral domain? A field?
      
      

Extra Credit

1. Let \( R \) be a ring and let \( I \) and \( J \) be ideals in \( R \).
   
   
   1. Prove that \( I+J = \{ x+y \,|\, x \in I \mbox{ and } y \in j \} \) is an ideal of \( R \).
      
      
   2. Prove that \( I \cap J \) is an ideal of \( R \).
      
      
   3. Prove that \( IJ = \{ x_1y_1+\cdots+x_ky_k \,|\, x_i \in I \mathrm{\;and\;} y_i \in J \} \) is an ideal of \( R \).
      
      
   4. Let \( x,y \in R \) where \( R \) is a commutative ring. Show \( (x) \subseteq (y) \) if and only if \( y \) divides \( x \) in \( R \).
      
      
   5. Consider the principle ideals \( (4) \) and \( (6) \) in \( \mathbb{Z} \). What ideal do we get when we add them together: \( (4)+(6) \)? Intersect: \( (4) \cap (6) \)? Multiply \( (4)(6) \)?
      
      
   6. Make a conjecture about the relationship between \(m,n,d,\ell,p \in \mathbb{Z} \) if as ideals we have "\( (m)+(n) = (d) \), \( (m) \cap (n) = (\ell) \), and \( (m)(n)=(p) \)".
      
      
2. Let \( R \) and \( S \) be rings with 1. Let \( \mathbb{F} \) be a field.
   
   
   1. Show \( I=R \) if and only if \( I \) contains a unit.
      
      
   2. Let \( \mathbb{F} \) be a field. List all of the ideals of \( \mathbb{F} \).
      
      
   3. Let \( \varphi:\mathbb{F} \to S \) be a ring homomorphism where \( \mathbb{F} \) is a field. Show that either \( \varphi(x)=0 \) for all \( x \in \mathbb{F} \) or \( \varphi \) is one-to-one.
      
      
3. Let \( I \) and \( J \) be ideals of some ring \( R \). We already know \( I+J \) is an ideal of \( R \).
   
   
   1. Briefly explain why \( J \) is an ideal of \( I+J \). [Thus justifying our use of \( (I+J)/J \) in the next part of the problem.]
      
      
   2. Prove \( I/(I \cap J) \cong (I+J)/J \). Hint: Use a projection homomorphism from \( I \) onto \( (I+J)/J \). Then use the first isomorphism theorem.
      
      
4. A couple of quotients.
   
   
   1. Let \( I = (x^2) \). Create addition and multiplication tables for \( \mathbb{Z}_2[x]/I \). Is this quotient ring a field?
      
      
   2. Let \( I = (x^3+x+1) \). Create addition and multiplication tables for \( \mathbb{Z}_2[x]/I \). Is this quotient ring a field?