Math 520B
Written Assignment No. 3

due Monday, October 31, 2005

Directions. Although you may refer to books for definitions and standard theorems, searching for solutions to these written exercises is not permitted. You may not seek help from others.

Bear in mind that rings are always assumed to have a multiplicative identity, and a homomorphism of rings is always assumed to carry the multiplicative identity of its domain to that of its target. Recall that if T is a ring, the term T-algebra indicates, by definition, a pair

<R, rho>

where R is a ring and rho : T --> R is a ring homomorphism.

  1. If F is a field and n > 0 an integer, it has been explained in this course that an F [t]-module structure on F^{n} extending its usual F-module structure amounts to the same thing as an n \times n matrix in F. Show that the F [t]-modules corresponding to two given n \times n matrices A and B are isomorphic if and only if A and B are similar over F.

  2. For a prime p > 1 it is a fact that the sequence of coefficients in the p-adic expansion of a p-adic integer that is rational must eventually repeat.

    1. Find the 5-adic expansion of the rational number 2/3.

    2. Find the expansion of 2/3 as a real “decimal” computed in base 5.

  3. If T is a ring, employ a suitable universal mapping property to state what is meant by an abstract product of a non-empty collection of T-algebras. Then prove that such always exist.

  4. Employ a suitable universal mapping property to state what is meant by a direct limit for a non-empty collection of rings indexed by a directed set with a suitable collection of connecting ring homomorphisms. Use this definition in carrying out the following task.

    Let A be a domain, K its field of fractions, and P a prime ideal in A. Let A_{P} denote the subring of K consisting of those elements of K representable as a/b where a, b \in A with b \inP, and for f \in A let A_{f} denote the subring of K consisting of those elements of K representable as a/f^{n} for some integer n > 0. Note that when g is a multiple of f, there is an inclusion homomorphism from A_{f} to A_{g}. Show that A_{P} is a direct limit of {A_{f} | f \inP}.

  5. Let F be a field, V, W finite-dimensional vector spaces over F, and \v{V} the dual of V. Prove that

     T  =  Hom_{F}(\v{V}, W) 

    together with the bilinear map tau : V \times W --> T defined by

     (v, w)  :--->  
    [phi  :-->  phi(v) · w]
    		 

    is a tensor product of V and W over F.