On quasi divisor theories and systems of valuations

Abstract

Abstract. We study the relationship between divisor theories and systems of valuations, and characterize monoids with quasi divisor theories of finite character by systems of essential valuations. Throughout, we avoid ideal theory but use divisor theoreticai methods.

Bases of Pre-Riesz Groups and Conrad’s F-Condition

“…On the other hand, P. Jaffard and J. Ohm generalized Krull's result to show that for a given l.o. Abelian group G, there exists a Bézout domain D such that the group of divisibility G(D) is isomorphic to G. Note that in the context of monoids and (generalized) divisor theories, Geroldinger and Halter-Koch got a nice proof of the Jaffard-Ohm correspondence in [12]. Furthermore, Rump and Yang [17] gave a categorical interpretation of the Jaffard-Ohm correspondence and established a general extension theorem for valuations with values in an abelian l-group, which yields a proof of Anderson's conjectural refinement of the Jaffard-Ohm theorem.…”

Hereditary atomicity in integral domains

“…It is important to note that nonnegative elements under this partial ordering are precisely the cosets that are represented by elements of R. In the 1960s and 1970s, there was a flurry of interest in the group of divisibility (see, for example, [62,63]), and perhaps the most well-known result of this time was the Jaffard-Ohm-Kaplansky Theorem (sometimes referred to as the Krull-Kaplansky-Jaffard-Ohm Theorem). We record this celebrated theorem here, and the interested reader can find a short divisor-theoretical proof of the same in [39,Corollary 2].…”