α-prime ideals

Let r be a commutative ring with identity. we give a new generalization to prime ideals called α-prime ideal. a proper ideal p of r is called an α-prime ideal if for all a, b in r with ab ∈ p, then a ∈ p or α(b) ∈ p where α ∈ end(r). we study some properties of α-prime ideals analogous to prime ideals. we give some characterizations for such generalization and we prove that the intersection of all α-primes in a ring r is the set of all α-nilpotent elements in r. finally, we give new versions of some famous theorems about prime ideals including α-integral domains and α-fields.