A true relative of Suslin's normality theorem

TL;DR

This paper proves a normality theorem for true elementary subgroups of SL_n(A), extending Suslin's classical results and significantly generalizing Mennicke's theorem for algebraic structures over rings.

Contribution

It introduces a normality theorem for true elementary subgroups, broadening the scope of Suslin's and Mennicke's foundational results.

Findings

Normality of true elementary subgroups established

Generalization of Mennicke's theorem achieved

Extension of Suslin's normality theorem demonstrated

Abstract

We prove a normality theorem for the "true" elementary subgroups of $SL_n(A)$ defined by the ideals of a commutative unital ring $A$. Our result is an analogue of a normality theorem, due to Suslin, for the standard elementary subgroups, and it greatly generalizes a theorem of Mennicke.