Abstract:Let R be a ring and I(# R) a right ideal of R. The right ideal I is  a strongly prime right ideal if, 

for each pair of  a and b in R,  and  imply that either  or ,and we call I a strongly 

semi-prime right ideal whenever  and  imply that . 
 
                In this work, we obtain three main results. In the first one we prove that the following are equivalent. 

               (i) R is strongly regular.(ii)R  is regular and every prime right ideal is strongly prime.(iii)R  is right weakly 

regular and every maximal right ideal is a two sided ideal. (iv)R  is semi-prime, every prime right ideal is strongly 

prime and R / P is regular for every completely prime ideal P of R .(v)R  is right weakly regular, left s-unital 

AC-ring and every prime right ideal is strongly prime.(vi) Any right ideal (except R  itself) is strongly semi-prime. 
 
               The second result shows that, if R is a ring with identity, then the right R -module R / I has a projective 

cover if and only if there is a non-zero idempotent \ I such that  eI  is small under the assumption that 

I is a strongly prime right ideal,where N(I) is the largest subring of R  containing  I. 

                 The third result reveals that a projective strongly prime right ideal in a self  injective regular ring with 

identity is a direct summand.The last two properties generalize Koh's and Karamzadeh's results, respectively. 

Key words: Strongly Prime (strongly semi-prime) right ideal, strongly (right weakly) regular ring, AC-ring, 

projective cover.