**189-571B:** Higher Algebra II

## Assignment 3

## To be presented: Wednesday, January 31.

The problem marked with a (*) is for everyone to do and to hand in on
the day of the problem session.
I will return this problem to you on Friday of the same week,
so late problems will not be accepted!

If you are stuck on your problem, come see me during my Monday
office hours (or make an appointment)
to get a hint...

**1**. (*)
Let N be a positive integer, and let R=Z/NZ be the ring of residue
classes modulo N.

a) Give a necessary and sufficient condition on
N for R to be primitive.

b) Give a necessary and sufficient condition on
N for R to be semi-primitive.

c) What is the Jacobson radical of R?

**2**. ** (Marni Mishna) **
Let R be a ring containing a nonzero ideal N that is
nilpotent, in the sense that N^{m}=0 for some integer m.
Show that R is not semi-primitive.

**3**. ** (Christian Cote)**
Show that if a ring R is primitive, then so is the
ring M_{n}(R) of nx n matrices with entries in R.