we have a sears deadbolt that is installed without screws.  How is this removed?

Openning the door, you don't see screws top and bottom of the bolt?

Thank you for your response.  There are no screws anywhere on the
deadbolt, the retaining ring is not threaded on and the inside plate
that holds the key does not turn either.

I looked at sears.com, and they are suspiciously lacking in door
locks... If there are no screws on the metal plate in the door where
the bolt comes through the hole, how is that plate fastened to the
door? This is one strange deadbolt.
Also, look for a small gap near the bottom of the ring around the
inside surface of the door. You might be able to pry/pop the ring off
with a flat tip screwdriver.

Thank you too for getting back with me.  This lock is most likely
about 20 years old.  There is no notch, and I tried to pry the
retaining ring off first and that did not work.  I contacted Sears
service and they told me that they only give service and advice on
major appliances only.

Time to call a locksmith :).

You might try turning the plate a tiny bit and gently prying each time
you turn it and see if this works. The plate may have a tab on the
inside that must be lined up with a notch before it can be popped off.

tutuzdad-ga

In other words, the notch and tab may be INSIDE and not visible.

Thanks tutuzdad, I will try turning and prying at the same time. 
Otherwise it is get the saw out or replace the door!!!

Maybe not. The only thing I know of that's put on and not intended to
EVER be removed is a tattoo. There's got to be a way. It's figuring it
out that's the trick.

Dad

Is the door locked? Do you have the key?