# Some Commutative Algebra

# Some Commutative Algebra

This chapter provides a background on commutative algebra and gives a self-contained proof of Johnson's Theorem 5.9.1 on regular solutions of systems of algebraic differential equations. It presents the facts on regular local rings and Kähler differentials needed for Theorem 5.9.1. It also recalls a common notational convention concerning a commutative ring *R* and an *R*-module *M*, with *U* and *V* as additive subgroups of *R* and *M*. Other topics include the Zariski topology, noetherian rings and spaces, rings and modules of finite length, integral extensions and integrally closed domains, Krull's Principal Ideal Theorem, differentials, and derivations on field extensions.

*Keywords:*
commutative algebra, Johnson's Theorem, algebraic differential equation, regular local ring, Kähler differentials, commutative ring, Zariski topology, noetherian ring, integrally closed domain, Krull's Principal Ideal Theorem

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