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Wednesday, June 19, 2013

Nonsimple Tensors (10.4.20)

Dummit and Foote Abstract Algebra, section 10.4, exercise 20:

MathJax TeX Test Page Let R=Z[x] and let I=(2,x). Show that the element 22+xx in IRI is not a simple tensor.

Proof: Define a mapI×IR(i,j)ijThis is clearly seen to be R-balanced, and so induces a homomorphism Φ on IRI. Assuming 22+xx=ab, then alsoΦ(22+xx)=Φ(22)+Φ(xx)=x2+4=Φ(ab)=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 abII. 

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