{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

MA554_AUG03

# MA554_AUG03 - QUALIFYING EXAM Math 554 August 2003 1(12...

This preview shows pages 1–2. Sign up to view the full content.

QUALIFYING EXAM Math 554 August 2003 1. (12 points) Without proof give an example of: (a) a domain R and a finitely generated torsionfree R -module that is not free; (b) a nonzero commutative ring R that is not a field so that every R -module is torsionfree; (c) a normal matrix with entries in R that is not similar to a diagonal matrix with entries in R . 2. (13 points) For R a commutative ring and M an R -module let M ? = Hom R ( M, R ) denote the R -dual of M . (a) Let M and N be R -modules. Show that ( M N ) ? = M ? N ? . (b) Let R be a principal ideal domain and M a finitely generated R -module. Show that M ? is a free R -module with rank M ? = rank M . 3. (12 points) Let Q [ X ] be the polynomial ring in one variable over Q , n a positive integer, and R = Q [ X ] / ( X n ). Classify all finitely generated R -modules up to R -isomorphisms. 4. (15 points) Let R be a commutative ring, F a free R -module of finite rank, and ϕ End R ( F ) an R -endomorphism of F . Show that the following are equivalent: (a) ϕ is bijective; (b) ϕ is surjective; (c) det( ϕ ) is a unit of R . 5. (20 points) Let K be a field, V a K -vector space of dimension n , End K ( V ) the K -vector space of K -endomorphisms of V , and ϕ End K ( V ) a fixed K - endomorphism. Show that:

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 2

MA554_AUG03 - QUALIFYING EXAM Math 554 August 2003 1(12...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online