Let R be a Noetherian domain with quotient field K and let be an algebraic field extension of degree d over K. If a is an anti-integral element of degree d over for each prime ideal p of depth one in R, then a is an anti-integral element of degree d over R. Moreover, is an anti-integral and flat extension over for each prime ideal p of depth one in R and is an algebraic field extension of degree d over K for then b is an anti-integral element of degree d over R.