On Prime One-Sided Ideals

Let R be a ring and let Lγ(R) be the lattice of right ideals. We define that I ∊ Lγ(R) is a prime right ideal provided that if AB⊆I for some A, B in Lγ(R) such that AI⊆I then either A⊆I or B⊆I. Any prime ideal of a ring R is a prime right ideal and if R is commutative then an ideal is prime if and only if it is a prime right ideal. If R is a ring and a∊R, let aR={x ∊ R | x=ar for some r∊R} and aR1={x ∊ R | x = na+ar for some integer n and r ∊ R}.