The Ohm-Rush content function

TL;DR

This paper explores the properties of the Ohm-Rush content function in ring extensions, introduces a new semicontent algebra concept, and compares these with classical content notions, revealing transitivity and equivalences in power series contexts.

Contribution

It introduces the semicontent algebra property, analyzes transitivity of content-related properties, and compares Ohm-Rush content with classical content in power series over valuation rings.

Findings

Gaussian, weak content, and semicontent properties are transitive.

Transitivity of content algebra property remains unknown.

Content properties coincide for power series over valuation rings with certain value group structures.

Abstract

The content of a polynomial over a ring $R$ is a well understood notion. Ohm and Rush generalized this concept of a content map to an arbitrary ring extension of $R$, although it can behave quite badly. We examine five properties an algebra may have with respect to this function -- content algebra, weak content algebra, semicontent algebra (our own definition), Gaussian algebra, and Ohm-Rush algebra. We show that the Gaussian, weak content, and semicontent algebra properties are all transitive. However, transitivity is unknown for the content algebra property. We then compare the Ohm-Rush notion with the more usual notion of content in the power series context. We show that many of the given properties coincide for the power series extension map over a valuation ring of finite dimension, and that they are equivalent to the value group being order-isomorphic to the integers or the reals. Along the way, we give a new characterization of Pr\"ufer domains.