Solving simultaneous congruences and decomposing rings into direct products. Problem-Solving Strategies
"Prove ( \mathbbZ[x] / (x) \cong \mathbbZ )." Solution narrative: Define ( \phi: \mathbbZ[x] \to \mathbbZ ) by ( \phi(f(x)) = f(0) ). This is a homomorphism, surjective, kernel = polynomials with constant term 0 = ( (x) ). By FIT, ( \mathbbZ[x] / (x) \cong \mathbbZ ). dummit and foote solutions chapter 7