Example. splitting the Group Algebra of C₂ [fgap-0011]
Example. splitting the Group Algebra of C₂ [fgap-0011]
Let \(C_2=\{1,s\}\) with \(s^2=1\), and suppose 2 is invertible in the nonzero commutative ring \(R\). The basis element \([s]\) is central, and \[ e_+=\frac {[1]+[s]}2,\qquad e_-=\frac {[1]-[s]}2. \] The algebra-product theorem becomes \[ R[C_2]\cong R\times R. \]
The isomorphism and its inverse are explicit: \[ \begin {aligned} a[1]+b[s]&\longmapsto (a+b,a-b),\\ (x,y)&\longmapsto \frac {x+y}{2}[1]+\frac {x-y}{2}[s]. \end {aligned} \] Under this map, \(e_+\) goes to \((1,0)\) and \(e_-\) goes to \((0,1)\). Thus the two idempotents are the coordinate projections, not merely a dimension count.