When a socket is wired bya radial method using 1 live and 1 neutral and 1 earth the circuit is already broken and is only completed when a load is used by inserting the plug and switching on.
When a second socket is needed the live/neutral/earth wires are taken from those entering the first socket
So now you have the original circuit and a extended circuit and a third socket would give another extended circuit and so on until you reach the last socket.
Depending which socket you remove the wire from will decide which sockets are no longer capable of making a circuit when a load is used.
A break in either the live or the neutral will have the same effect.
A ring main method is wired in the same manner, but instead of finishing at the last socket the cable containing the live/neutral/earth is continued back to the fusebox to the same terminals as the the original outgoing cable.
So you have one cable going clockwise and another going anticlockwise feeding the sockets, the reason being they share the load and by comparison a smaller cable can be used
Now if you removed a wire or in fact the length of cable between two sockets, the sockets would still function but not as a ring.
You will have created two radial circuits
In practice you may not have a true 'ring main', Any spurs taken from a socket or junction box are radial
Hope this helps to clarify the position.
unless I've got it wrong