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Regular elements of a ring, monic polynomials and “lcm-coprimality”
Darij Grinberg May 22, 2021
P5
2. Regular elements (a.k.a. non-zero-divisors)
2.1. Definition
We begin with a basic notation:
Definition 2.1. Let A be a commutative ring. Let a ∈ A.
The element a of A is said to be regular if and only if every x ∈ A satisfying ax = 0 satisfies x = 0.
Instead of saying that a is regular, one can also say that “a is cancellable”, or that “a is a non-zero-divisor”.
This notion of “regular” elements has nothing to do with various other notions of “regularity” in commutative algebra (for example, it is completely unrelated to the notion of a “von Neumann regular element” of a ring).
It might sound like a bad idea to employ a word like “regular” that has already seen so much different uses; however, we are not really adding a new conflicting meaning for this word, because the word is already being used in this meaning by various authors (among them, the authors of [LLPT95]), and because our use of “regular” is closely related to the standard notion of a “regular sequence” in a commutative ring 4.
Many authors (for example, Knapp in [Knapp2016]) define a zero divisor in a commutative ring A to be a nonzero element of A that is not regular.5 Thus, at
least in classical logic, regular elements are the same as elements that are not zero divisors (with the possible exception of 0). I find the notion of a “zero divisor”less natural than that of a regular element (it is the regular elements, not the zero divisors, that usually exhibit the nicer behavior), and it is much less suitable for constructive logic (as it muddies the waters with an unnecessary negation), but it appears to be more popular for traditional reasons.
(引用終り)
以上
結論：用語"cancellable"の意味が理解できないおサルさんでしたとさ