Abstarct: Let R be a constant ring which is α-compatible for an endomorphism α of R and f(x)=a_0+a_1 x+⋯+ a_n x^n∈R[x;α] be a nonzero skew polynomial in R[x;α]. It is proved that if there exist a nonzero skew polynomial g(x)=b_0+b_1 x+⋯+ b_m x^m in R[x;α] such that g(x) f(x)=c is a constant in R, then b_0 a_0=c and there exist nonzero elements a and r in R such that rf(x)=ac. In particular, r=ab_p for some p, 0≤p≤m, and a is either one or a product of at most m coefficients from f(x). Furthermore, if b_0 is a unit in R, then a_1 , a_2 , … , a_n are all nilpotent. As an application of above result, it is proved that if R is weakly 2-primal ring which is α-compatible for an endomorphism α of R, then a skew polynomial f(x) in R[x;α] is a unit if and only if its constant term is unit in R and other coefficients are all nilpotent. Moreover, for an endomorphism α and α-derivation δ, we show that if R is (α,δ)-compatible then R is 2-primal if and only if the Ore extension R[x;α,δ] is 2-primal if and only if N(R)=N_* (R;α,δ) if and only if N(R)[x;α,δ]=N_* (R[x;α,δ]) if and only if every minimal (α,δ)-prime ideal of R is completely prime. Also, we show that the semiprimeness, primeness and reducedness can go up to unique product monoids (simply, u.p-monoids). By these results we can compute the lower nil-radical of u.p-monoid rings, from which the well-known fact of Amitsur and McCoy for the polynomial rings can be extended to u.p-monoid rings. Finally, we show that if R be a reversible ring which is δ-compatible for a derivation endomorphism δ on R and f(x)=a_0+a_1 x+⋯+ a_n x^n be a nonzero differential polynomial in R[x;δ] that if there exists a nonzero polynomial (x)=b_0+b_1 x+⋯+ b_m x^m in R[x;δ] such that g(x) f(x)=c be a constant in R, then b_0 a_0=c and there exists a nonzero elements a and r in R such that rf(x)=ac. Furthermore, we show that if b_0 is unit in R, then a_1 , a_2 , … , a_n are all nilpotent.