"Rzecz" can be used in sense of "przedmiot" and actually it is used more often than "przedmiot" but it can be also used in abstract sense. For example the official name of Poland in Polish language is "Rzeczpospolita Polska". Where "Rzeczpospolita" is a literal translation of Latin res publica (Rzecz + pospolita) and means "Republic".
Przedmiot can also be used in somehow abstract sense(cf. przedmiot nauczania) and I'm not so sure about frequency of use between the two.
Also, while official title of Poland in English is indeed Republic of Poland, "rzeczpospolita" on it's own is better translated as commonwealth(republic is "republika" in Polish). That is also how Rzeczpospolita Obojga Narodów is/was usually translated into English(Polish–Lithuanian Commonwealth). ;)