If P is an ideal in a not necessarily commutative ring R, then the following conditions are equivalent.

(a) P is a prime ideal.

(b) If r,s ϵ R are such that rRs ⊂ P, then r ϵ P or s ϵ P.

(c) If (r) and (s) are principal ideals of R such that (r)(s) ⊂ P, then r ϵ P or s ϵ P.

(d) If U and V are right ideals in R such that UV ⊂ P, then U ⊂ P or V ⊂ P.

(e) If U and V are left ideals in R such that UV ⊂ P, then U ⊂ P or V ⊂ P.