A noninteracting electron gas on a one-dimensional ring is considered at finite temperatures. The localized spin is embedded at some point on the ring, and it is assumed that the interaction between this spin and the electrons is the exchange interaction, being the basis of the Ruderman-Kittel-Kasuya-Yosida indirect exchange effect. When the number of electrons is large enough, it turns out that any small but finite interaction radius value can always produce an essential change in the spin density oscillations in comparison with the zero interaction radius traditionally used to model the Ruderman-Kittel-Kasuya-Yosida effect.