ring_properties

std.algebra.ring_properties · Level L3

Three properties of a finite ring with 1, each checked over every pair: whether it is a commutative ring (a ring with ab = ba for all a, b), as 1 or 0; its zero divisors, the nonzero a with ab = 0 or ba = 0 for some nonzero b, as a 0/1 mask; and its units, the a with a two-sided inverse, ab = ba = 1, as a 0/1 mask.

ab = ba; a ≠ 0, ab = 0; ab = ba = 1

Signature

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

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 every pair of the multiplication table in exact rational arithmetic, on all 40 test cases.
All 300 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

Called by

is_field is_integral_domain

sha256:2856699627d3f597c8d9c428549a17d3c2cffa03a09d63a83a767d03d292e1f1

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.ring_properties · Ω:ring_properties
Pins: sha256:73f9eed31a759cde6979da3a8d78b08229b8e9a8b933452effbb3d980923ce6athis graph and the 4 it reaches through calls
Evidence: sha256:d23418d934f49d086e883988e1b95d5324c4fe370ef536051c3ab0de36316b56the 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.