$\mathbb ZS_n$-modules and polynomial identities with integer coefficients

TL;DR

This paper explores polynomial identities in rings over integers using $ ext{ZS}_n$-modules, deriving new results on codimensions and bases for specific algebra classes, and confirming conjectures in torsion-free rings.

Contribution

It introduces a reduction of polynomial identity study to proper identities for rings over integers and computes codimensions and bases for specific algebra classes.

Findings

Derived series of $ ext{ZS}_n$-submodules using Young's rule.

Calculated codimensions and bases for upper triangular matrices and Grassmann algebras.

Confirmed analogs of Amitsur's and Regev's conjectures in torsion-free rings.

Abstract

We show that, like in the case of algebras over fields, the study of multilinear polynomial identities of unitary rings can be reduced to the study of proper polynomial identities. In particular, the factors of series of $\mathbb ZS_n$-submodules in the $\mathbb ZS_n$-modules of multilinear polynomial functions can be derived by the analog of Young's (or Pieri's) rule from the factors of series in the corresponding $\mathbb ZS_n$-modules of proper polynomial functions. As an application, we calculate the codimensions and a basis of multilinear polynomial identities of unitary rings of upper triangular $2\times 2$ matrices and infinitely generated Grassmann algebras over unitary rings. In addition, we calculate the factors of series of $\mathbb ZS_n$-submodules for these algebras. Also we establish relations between codimensions of rings and codimensions of algebras and show that the analog of Amitsur's conjecture holds in all torsion-free rings, and all torsion-free rings with 1 satisfy the analog of Regev's conjecture.