Ecole Polytechnique Federale de LausanneDepartement de mathematiques

Prof. E. Bayer Fluckiger

Algebra for digital communicationsSection de syste`mes de communication winter semester 2004-2005

Exercise sheet 4 9.11.04

Exercise 1Let R be a ring and x R. Suppose there exists a positive integer n such that xn = 0. Showthat 1 + x is a unit, and so is 1 x.Exercise 2Consider the following elements.

1. 3 Z/21Z.2. 5 Z/21Z.3. 5 Z/25Z.4. 12 Z/23Z.

For each of them, decide if it is unit. If yes, give its inverse. If not, show explicitly that it is azero divisor.

Exercise 3Let A,B be two rings, and let f : A B be a ring homomorphism. Show that the image of fIm(f) is a subring of B.

Exercise 4Let n, m Z such that m|n. Consider the map

f : Z/nZ Z/mZwhich maps an element of Z/nZ to its class modulo m.

1. Show that the map f is well defined (i.e. the image of f does not depend on the represen-tative).

2. Show that f is a ring homomorphism. Is it onto ?

3. Compute the kernel Ker(f) of f . In particular, what is its cardinality ? Is f one-to-one ?

4. Show that the image of a unit of Z/nZ is a unit of Z/mZ.5. What can we say if we remove the condition m|n ? Is there always such a ring homomor-

phism g : Z/nZ Z/mZ ?Challenge Exercise 1Let R be a ring, and let a 6= 0 R such that there exists an integer n with an = 0. Show thatR (R[X]) and R 6= R[X], where R and R[X] denote respectively the group of units ofR and R[X].