This is a write-up of lectures given at the "Kleine Herbstschule 9" of the Graduiertenkolleg "Mathematik im Bereich Ihrer Wechselwirkungen mit der Physik" at the Ludwig-Maximilians-Universitaet München. Starting from classical algebraic geometry over the complex numbers (as it can be found for example in Griffiths and Harris it was the goal of these lectures to introduce some concepts of the modern point of view in algebraic geometry. Of course, it was quite impossible even to give an introduction to the whole subject in such a limited time. For this reason the lectures and now the write-up concentrate on the substitution of the concept of classical points by the notion of ideals and homomorphisms of algebras.
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