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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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The total quotient ring of a nondomain need not be a field:
Statement refuted
The total quotient ring of every nonzero commutative ring is a field.
Facts & Assumptions
Given: The quotient ring . By definition, a regular element has trivial annihilator, and the total quotient ring is the localisation at all regular elements.
Units form a group, and a map already taking all denominators to units extends uniquely through localisation (The units of a ring are the invertible elements of its multiplicative monoid, and is a group under multiplication; only in the zero ring, Universal property of localisation: maps that invert factor uniquely through ).
The ring has congruence-class arithmetic (For every , the congruence-class ring is the quotient ring ).
In a field, every nonzero element is a unit (Field).
Counterexample
In , the class is not regular because it annihilates ; the classes are nonzero zero divisors; and are units and hence regular. Thus the regular elements are exactly .
Since the identity map of already sends every element of to a unit, [F1] gives an inverse to the localisation map, so .
The nonzero class of is not a unit because every product is even modulo and cannot equal . Hence , and therefore , is not a field by [F3].
Depends on
- Universal property of localisation: maps that invert $S$ factor uniquely through $S^{-1}R$
- The units of a ring are the invertible elements of its multiplicative monoid, and $R^{\times}$ is a group under multiplication; $0 \in R^{\times}$ only in the zero ring
- For every $n\in\mathbb N$, the congruence-class ring $\mathbb Z/n$ is the quotient ring $\mathbb Z/n\mathbb Z$
- Field
Used by
Nothing in the library uses this result yet.
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 49 results over 15 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- The Stacks Project, Section 10.25: Zerodivisors and total rings of fractions (standard reference, not scraped)