For any integral domain , the group of invertible matrices with coefficients in is the amalgamated free product of invertible matrices over and invertible upper triangular matrices over , amalgamated over the upper triangular matrices of . More compactly
![$ k[t]$](http://images.planetmath.org:8080/cache/objects/4747/l2h/img3.png "$ k[t]$"){kind=small}
![$ k[t]$](http://images.planetmath.org:8080/cache/objects/4747/l2h/img5.png "$ k[t]$"){kind=small}
![$\displaystyle \mathrm{GL}_{n} (k[t])\cong\mathrm{GL}_{n} (k) *_{B(k)}B(k[t]).$](http://images.planetmath.org:8080/cache/objects/4747/l2h/img7.png "$\displaystyle \mathrm{GL}_{n} (k[t])\cong\mathrm{GL}_{n} (k) *_{B(k)}B(k[t]).$"){kind=small}