Why do some married women keep their maiden names and do not wear a wedding ring? If you meet someone like that and begin a conversation with her and ask her to have a drink, maybe she will say I'm sorry, I'm a married woman. Then why not show it?

I'm sure in a good number she is not married and is just trying to tell you nicely to leave her alone.

Some women hate the whole marriage thing and therefore reject the trappings of marriage like taking the man's last name and wearing the ring, even though they decide to get married for whatever reasons.  I know a few women who are married and follow that path.

Then again there are married women who cruise for affairs and so don't wear rings while doing so.

I'm sure there are other reasons as well.

BUT some do not at work because the ring gets in the way (say cooking or similar tasks).  I stopped wearing my ring when I was working with machines...  a relative nearly lost their hand when a ring got caught in machinery.  
A lot of women keep a name to match their children's name or continue to use that name after remarriage or hyphenate.  Or continue to use the name they stated their career with even though married...  some use that name professionally & a married name off the job.  
Look at actresses who commonly keep one name professionally...  no telling who they are married to at the time.    
I don't have a solution...   If they decline an invitation, glance at their hands & they say sorry, I didn't know.