In this paper we study certain general properties of primitive and 0-primitive idempotents, and the obtained results we apply to 0-inversive E-inversive and B-inversive semigroups. We also determine some conditions under which a left ideal of a semigroup with zero generated by a nonzero idempotent is left 0-simple. The obtained results generalize some results by Venkatesan, Steinfeld, Bogdanović and Milić, Bogdanović and Ćirić, Mitsch and Petrich and others.