关于环中非零非单位元、可约元及合成元的定义咨询

Hey Jens, great question—formalizing Unique Factorization Domains (UFDs) can definitely throw up terminology inconsistencies, especially since different algebraic texts use slightly varying language. Let me break this down clearly for you:

First off, the standard, widely accepted term for a ring element that’s neither zero nor a unit is simply non-zero non-unit element. Some texts might shorthand this in context, but this full phrase is unambiguous and used across most modern algebra references.

Here’s where things can get a bit nuanced—let’s start with the most consistent definitions:

Reducible Elements #

In an integral domain ( R ), a non-zero non-unit element ( a \in R ) is reducible if there exist non-zero non-unit elements ( b, c \in R ) such that ( a = bc ).

The complement of this is an irreducible element: a non-zero non-unit that cannot be written as a product of two non-zero non-units. This is the core distinction you’ll need for UFDs, since UFDs require every non-zero non-unit to factor uniquely into irreducibles.

Composite Elements #

Terminology here varies more:

• Most modern texts (like Dummit & Foote, Fraleigh) use composite element as a direct synonym for reducible. In these sources, "composite" is just an alternative word for the same definition above.
• A smaller number of older or more specialized texts (e.g., Zariski & Samuel’s Commutative Algebra) draw a subtle distinction: they define a composite element as a non-zero non-unit that can be written as a product of at least two irreducible elements (not just any non-units). Under this stricter definition, a reducible element that factors into other reducibles (but not yet irreducibles) wouldn’t be called composite until it’s fully decomposed into irreducibles.

For formalization work, sticking to widely accepted, modern references will minimize ambiguity:

• Algebra: Chapter 0 by Paolo Aluffi: Uses "reducible" exclusively for the product-of-non-units definition, and avoids "composite" to prevent confusion.
• Abstract Algebra by Dummit & Foote: Clearly defines reducible elements, notes that "composite" is sometimes used synonymously, and emphasizes irreducibility as the key concept for UFDs.
• A First Course in Abstract Algebra by John Fraleigh: Uses both "reducible" and "composite" interchangeably, with the standard product-of-non-units definition.
• Commutative Algebra by Zariski & Samuel: Includes the stricter composite-as-product-of-irreducibles definition if you need to account for niche terminology.

If you’re formalizing this in a proof assistant (like Coq, Lean, or Isabelle), I’d recommend explicitly defining your terms upfront using the modern reducible/irreducible distinction—this aligns with most existing formalizations of UFDs.

内容的提问来源于stack exchange，提问作者Jens Wagemaker