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Absence of meromorphic identities in M_d(R) (extension of Amitsur’s theorem)

Determine whether there exist non-trivial meromorphic identities in M_d(R), R ∈ (0,+∞]; equivalently, establish an extension of Amitsur’s theorem to the skew fields M_d(R) asserting that any element that evaluates to 0 everywhere it is defined on the non-commutative row-ball must be the zero element.

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Background

For the free skew field, Amitsur’s theorem asserts that rational identities do not exist: a non-commutative rational function that vanishes wherever defined is identically zero. The authors discuss potential meromorphic identities in M_d(R), which would undermine interpreting M_d(R) elements as NC functions.

They anticipate that an extension of Amitsur’s theorem to M_d(R) should hold but note that necessary additional results are not yet established, leaving the existence of meromorphic identities open.

References

However, there could exist, in prinicple, a non-zero element $0\neq f \in {M} _d (R)$, with $0 \in Dom f$, which evaluates identically to $0$, everywhere it is defined in $R \cdot B$. Although we expect that there should be an extension of Amitsur's theorem to ${M} _d (R)$, $R \in (0, +\infty]$, we have not yet been able to establish this.

— Rings of non-commutative functions and their fields of fractions (2509.21270 - Augat et al., 25 Sep 2025) in Section 5 (The universal skew field of fractions of O_d(R))