I don't know about rich/poor suicide rates but in developed/developing countries the suicide rates are higher than in undeveloped world.
http://en.wikipedia.org/wiki/List_of_countries_by_suicide_ra...
http://en.wikipedia.org/wiki/List_of_countries_by_suicide_ra...
From glancing at that chart, there also appear to be correlations with religion and distance from the equator (i.e. regions having more or less daily sunlight).