For "A" a commutative ring, one can define a determinant det: GL("A") → "A*" to the group of units of "A", which vanishes on E("A") and thus descends to a map det : "K"1("A") → "A*". As E("A") ◅ SL("A"), one can also define the special Whitehead group S"K"1("A") := SL("A")/E("A"). This map splits via the map "A*" → GL(1, "A") → "K"1("A") (unit in the upper left corner), and hence is onto, and has the special Whitehead group as kernel, yielding the split short exact sequence: