Math 421/621

Spring 2000

Homework 1

 

  1. Prove that if A is an R-algebra, then A is an R-module.

    2.  

  2. a) Let F be any field. Find all F-submodules of F and find all quotient modules of F.

    3. b) Do the same as in part a), replacing the field "F" with the integers Z.

     

  3. Let M, N, K be -modules. If and then Use this to show that is a ring with identity.

    4.  

  4. Let be a commutative ring with 1, show that as left modules.

    5.  

  5. Let be commutative with 1. Show that as rings. (G) See if you can find a counterexample to #10, pg. 331.

    6.  

  6. (G) Find where
denotes the real numbers. Characterize the elements that are isomorphisms.