is_integral_domain

std.algebra.is_integral_domain · Level L3

Whether a finite ring is an integral domain, as 1 or 0: a commutative ring with 1 ≠ 0 and no zero divisors, ab ≠ 0 for all nonzero a and b, checked over every pair. Every finite integral domain is a field (multiplication by a nonzero a is injective, so onto), but that is a theorem, not part of the definition.

commutative, 1 ≠ 0, ab = 0 ⇒ a = 0 ∨ b = 0

Signature

is_integral_domain(Ad: i64[n, n], Mu: i64[n, n]) → f64[]

Structure

The function as NOVA stores it: one box per input, operation and output, and arrows that carry values. A double border marks another library function this one runs — called once, or by Scan once per element; select it to open that function.

input
operation
constant
call
output

Verification

Signature proven by NOVA’s shape solver, for every size.
Equal to the reference a commutative ring with no zero divisor in exact rational arithmetic, on all 40 test cases.
All 40 float64 results are exact: the error is zero.
Interpreter and NumPy backend return bit-identical results.
At float32 too: the 40 test cases rounded to float32, and 40 of them exact, with zero error.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 100%
bit-equal to the NumPy formula in float64: 100%
largest error, in units in the last place: 0

Identity

Calls

ring_axioms ring_properties

Called by

—

sha256:c74824ab50829c26395eb85417e02a45622f3a88801cf305e8b21384dd288578

The semantic hash of the graph. It changes when the program changes, and never when only its documentation does.

Control handle

Symbol: Ω:std.algebra.is_integral_domain · Ω:is_integral_domain
Pins: sha256:1cc489bfc98733f715bd4fff133a2cecdb5effb6f8fdb7c8f4db4bec35598025this graph and the 5 it reaches through calls
Evidence: sha256:334b7b7198109b1a72ab53822dbb3d68003e8d012d5c95ef85882f723b74b7cathe hash of its verification record
Needs: no capability: a pure function

Through NOVA’s control layer, the symbol launches this function only while the program still matches what it pins: a change to this graph, or to any graph it reaches, needs a migration first.