Geodesic
An inverse matrix of an upper triangular matrix can be lower triangular
Discussiones Mathematicae. General Algebra and Applications, Tome 22 (2002) no. 2, pp. 161-166

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In this note we explain why the group of n×n upper triangular matrices is defined usually over commutative ring while the full general linear group is defined over any associative ring.
Keywords: upper tringular invertible matrix, group of matrices, Dedekind-finite ring 
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Hołubowski, Waldemar. An inverse matrix of an upper triangular matrix can be lower triangular. Discussiones Mathematicae. General Algebra and Applications, Tome 22 (2002) no. 2, pp. 161-166. http://geodesic.mathdoc.fr/item/DMGAA_2002_22_2_a5/