Baer Rings

Dear NG,

Apparently very little is being discussed about these rings in this NG,
so let me start =)

For starters.. the definition: (I use only commutative rings, but
everything can be adjusted to the noncommutative case)
A ring R is Baer iff for any ideal I in R, the annihilator of I :
I*={r in R : rI = (0)}
is generated by an idempotent in R, i.e. there is an e_I in R such that
e_I^2 = e_I and
I* = e_IR

Now.. there are some authors who would define a Rickart ring a Baer
ring. A Rickart ring is one with the same definition above, but instead
of *any I* we replace it by *any principal ideal I=aR*

Clearly Baer rings imply Rickart rings, but the converse is said to be
untrue .. which brings me to the question whether someone could provide
a very easy example. The author of the article in planetmath.org that
wrote the definition of Baer rings pointed out that the
"counterexample" can be found in the book of Lam, "Lectures on Modules
and Rings" (funnily theres another book "Lectures on Rings and
Modules")... It will take a few days before I get my hands on this
book, and before that I thought maybe someone could already make a nice
easy counterexample.. Im looking particularly on a counter example for
commutative unitary rings.

Sincerely,
Jose Capco

.

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