Determinantide

Determinantide is a term used in some mathematical discussions to denote a determinant-like invariant associated with square matrices and linear operators. It serves as a generalized notion of determinant, particularly in contexts where the ordinary determinant is not available or not unique, such as noncommutative rings or certain categories of modules.

Definition: In the standard setting of an n by n matrix A over a commutative ring R, determinantide(A) agrees with the ordinary determinant det(A). In more general settings, several constructions yield a scalar or scalar-like invariant that is described as determinantide; prominent examples include the Dieudonné determinant for matrices over division rings, and exterior-power-based invariants defined when the top exterior power can be identified with the base ring.

Properties: A determinantide aims to satisfy multiplicativity under suitable compositions, and normalization detide(I)=1. It is often required to be multilinear in columns and alternating in the commutative case. However, over noncommutative rings, no single universally accepted determinantide satisfies all desirable properties, so variants exist that relax linearity, normalize only on certain classes of matrices, or take values in a quotient group rather than the base ring.

Examples: For a commutative field, determinantide coincides with the usual determinant. For a division ring, the Dieudonné determinant provides a determinant-like invariant that is multiplicative but takes values in the abelianization of the multiplicative group, not necessarily the base ring itself.

Context: The term determinantide is not standardized in textbooks and is mostly encountered in expository discussions or as a convenience label for determinant-like constructions in noncommutative algebra and K-theory. Related concepts include the ordinary determinant, quasi-determinants, and reduced norms.