Let n be a positive integer. An f-ring A is said to satisfy the left nth-convexity property if for any u, v ∊ A such that v ≧ 0 and 0 ≧ u ≧ vn, there exists a w ∊ A such that u = wv. The right nth-convexity property is defined similarly and an f-ring is said to satisfy the nth-convexity property if it satisfies both the left and the right nth-convexity property. In this paper, we study arbitrary f-rings which satisfy one of the convexity properties.
Those f-rings which satisfy one or more of these properties have been studied by several authors. In [3, 1D], L. Gillman and M. Jerison note that any C(X) satisfies the nth-convexity property for all n ≧ 2, and in [3, 14.25], they give several properties that in C(X) are equivalent to the 1st-convexity property. M. Henriksen proves some results about the ideal theory of an f-ring satisfying the 2nd-convexity property in [5] and S. Steinberg studies left quotient rings of f-rings satisfying the left 1st-convexity property in [13].