rubber backed indoor carpet - what to go underneath?

I've just installed some custom-fit indoor carpetting in the upper landing (front door entrance) and lower landing (garage entrance) of our house.
there is vinyl flooring underneath it that is in decently good condition. I understand that within time, the rubber backing will react with the vinyl and turn it yellow....not something I want in the long run.
any suggestions as to what to put down on the floor to protect it? as the carpet has been cut to a custom-fit, having the carpet slipping on something is not an issue.
I had thought of an old bedsheet that was cut/folded appropriately...would that keep the rubber from reacting with the vinyl?
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Hamilton Audio wrote:

I do not know how "high class" the rubber on your carpet is, but I have had problems with the rubber on the back of bath mats sticking to the floor after a few years. I have no idea whether a sheet would be enough to
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