Ring with 4 elements

Let R be a unit ring with 4 elements. Let us prove that R is commutative. R ={0,1,a,b}. Let x,y be in R and let us prove that xy=yx. If x=0 then obviously xy=0=yx. If x=1 then xy=y=yx. We then have to prove that ab=ba. We have a+1∈ R. So a+1=0 or a+1=b. If a+1=0 then a=-1 and we have ab=(-1)b=-b=b(-1)=ba. If a+1=b then ab=a²+a=ba the thesis follows.