Proof: Define a mapI×I→R(i,j)↦ijThis is clearly seen to be R-balanced, and so induces a homomorphism Φ on I⊗RI. Assuming 2⊗2+x⊗x=a⊗b, then alsoΦ(2⊗2+x⊗x)=Φ(2⊗2)+Φ(x⊗x)=x2+4=Φ(a⊗b)=abSince x2+4 is a monomial quadratic with no roots, it does not factor in R, and thus either a or b is ±1 and now a⊗b∉I⊗I. ◻
No comments:
Post a Comment