Understanding tension force

I learnt in a physics class about tension where I was told that If we have string attached to a ceiling and other end attached to a block, then any cross sectional area of string divides it into two parts and the common magnitude of electromagnetic forces that these two parts exert on each other is called tension in string at that point.
Now if we have a string which is just kept on a floor. I was told that this string will have zero tension but what I doubt is that if still I consider a cross sectional area of string, the two parts on either side of this cross section are pulling each other as they are still in contact at that cross section and exert electromagnetic forces on each other therefore tension at that cross section must be non zero?