Math 3110 Spring 2010 -- Extra Credit Homework Problems

Due: Monday, April 26th
Turn in a few or as many as you like. These are optional problems for extra credit.
  1. Let \( R \) and \( S \) be rings. We can turn \( R \times S \) into a ring by defining operations "componentwise". Let \( (a,b), (x,y) \in R \times S \) and define \( (a,b)+(x,y)=(a+x,b+y) \) and \( (a,b)(x,y)=(ax,by) \). Notice \( a+x \) is addition in \( R \) and \( b+y \) is addition in \( S \). Likewise \( ax \) is multiplication in \( R \) and \( by \) is multiplication in \( S \).

    1. Prove that \( R \times S \) is a ring (you have to verify all of the axioms -- this isn't a subring of something else).

    2. Suppose \( R \) and \( S \) have 1's. Then show \( R \times S \) is also a ring with 1.

    3. Suppose that \( R \) and \( S \) are integral domains. Show that \( R \times S \) is not an integral domain.

  2. 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?

  3. 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. Consider the principle ideals \( (4) \) and \( (6) \) in \( \mathbb{Z} \). What ideal do we get when we add them together: \( (4)+(6) \)?

    3. Make a conjecture about the relationship between \(m,n,d \in \mathbb{Z} \) if as ideals we have "\( (m)+(n) = (d) \)".

  4. Let \( I \) and \( J \) be ideals of some ring \( R \). We already know (by the last problem) that \( I+J \) 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.