Locally Invariant and Semi-Invariant Right Cones

A right cone of a group G is a submonoid H of G so that for a, b ∈ H either aH ? bH or bH ? aH and G = {ab ^?1 | a, b ∈ H}. Valuation rings, right chain rings, the cones of right ordered groups provide examples. It is proved, see Theorem 17, that a semi-invariant right cone H with d.c.c. for prime ideals satisfies Ha ? aH for all a ∈ H, that is H is right invariant. Essential is the following Theorem 9: Let H be a locally invariant right cone in G with d.c.c. for prime ideals, and let I ≠ H be an ideal in H. Then P _l (I) ? P _r (I) for the associated left and right prime ideals of I.