mapping drives from a second networked computer

I have two computers networked through a router. One computer has one user account (computer A) and the other computer has four user accounts (computer B).

When I try to map a new network place from computer B to computer A the network can see the second computer B as part of the same workgroup but the computer cannot be accessed.

When I try to view workgroup computers and click on computer B the prompt to enter a user name and password pops up.

However, when computer B tries to access a shared folder on computer A no problem exists and no pop-up window appears either.

So in short what I would like to do is be able to access shared folders on computer B from computer A without having to enter a user name and password each time.

I do know this is possible because prior to reformatting the hd of computer A, a couple days ago, I was able to map new network places without having to see the login prompt appear. This would only be possible if the administrator of computer B was logged into computer B.

IF anyone knows how to do this it would be much appreciated.

Cheers,

CCOOL

What version of windows? And did you apply all the windows updates to computer A after you reformatted?

Sorry, I should have mentioned it in the post that I am running windows xp pro and I have installed the critical updates.