Convexity Conditions on <i>f</i>-Rings

Abstract

Let n be a positive integer. An f -ring A is said to satisfy the left n th -convexity property if for any u, v ∊ A such that v ≧ 0 and 0 ≧ u ≧ v n , there exists a w ∊ A such that u = wv . The right n th -convexity property is defined similarly and an f -ring is said to satisfy the n th -convexity property if it satisfies both the left and the right n th -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 n th -convexity property for all n ≧ 2, and in [ 3 , 14.25], they give several properties that in C(X) are equivalent to the 1 st -convexity property. M. Henriksen proves some results about the ideal theory of an f -ring satisfying the 2 nd -convexity property in [ 5 ] and S. Steinberg studies left quotient rings of f -rings satisfying the left 1 st -convexity property in [ 13 ].