Where does ideal collide?

In abstract algebra, an ideal is a subset of a ring that is closed under addition and that absorbs multiplication by any element of the ring, which is exactly the property needed to collapse the ring down to a well-defined quotient structure. Dedekind introduced ideals in his Theory of algebraic integers to restore unique factorisation in number systems where ordinary numbers fail to factor uniquely. The referent is a structural substructure of a ring that behaves like a target for quotienting, nothing to do with a standard of perfection or an analytical model of society.[1]

Unit

The set of ring elements absorbed under multiplication

Recovery means

Not applicable

Measured by

Characterised by its generators and by the quotient ring it produces (prime, maximal, principal, and so on)

In sociology, Max Weber's ideal type is a deliberately one-sided conceptual construct (an analytical exaggeration of a phenomenon's essential features, such as the 'ideal' bureaucracy or market) built not as a value judgment or an aspiration but as a measuring rod against which messy real cases can be described and compared. Weber set it out in Economy and society as a tool of interpretive social science. The referent is a methodological abstraction for comparison, unrelated to an algebraic substructure despite sharing the word 'ideal'.[2]

Unit

The gap between a real case and the pure constructed type

Recovery means

Not applicable

Measured by

Used qualitatively, by comparing concrete cases against the idealised construct's defining features