## Commutative Algebra 44

Fractional Ideals Throughout this article, let A be an integral domain and K its field of fractions. We do not assume the ring to be noetherian. The objective here is to develop the theory of multiplying and dividing certain classes of non-zero … Continue reading

## Commutative Algebra 43

Catenary Rings Let us look at prime chains in greater detail. Definition. Let be a chain of prime ideals of a ring A. We say the chain is saturated if for any prime ideal of A, ; maximal if it … Continue reading

## Commutative Algebra 42

Noether Normalization Theorem Throughout this article, k is a field, not necessarily algebraically closed. Definition. Let A be a finitely generated k-algebra which is an integral domain. We say are algebraically independent over k if they are so as elements … Continue reading

## Commutative Algebra 40

More on Integrality Lemma 1. Let be an integral extension. If is an ideal and , the resulting injection is an integral extension. Proof Any element of can be written as , . Then x satisfies a monic polynomial relation: . … Continue reading

## Commutative Algebra 39

Integrality Throughout this article, A is a subring of B; we will also call B a ring extension of A. Definition. An element is said to be integral over A if we can find (where ) such that in B. For example, is integral over since … Continue reading

## Commutative Algebra 38

Artinian Rings The main result we wish to prove is the following. Theorem. A ring A is artinian if and only if it is noetherian and , where denotes the Krull dimension. Note Recall that means all prime ideals of A … Continue reading

## Commutative Algebra 37

Artinian Modules Instead of the ascending chain condition, we can take its reverse. Definition. Let M be an A-module. Consider the set of submodules of M, ordered by inclusion, i.e. if and only if . We say M is artinian … Continue reading

## Commutative Algebra 36

In this article, we will study the topology of Spec A when A is noetherian. For starters, let us consider irreducible topological spaces in greater detail. Irreducible Spaces Recall that an irreducible topological space is a non-empty space X satisfying any of the … Continue reading

## Commutative Algebra 35

Noetherian Modules Through this article, A is a fixed ring. For the first two sections, all modules are over A. Recall that a submodule of a finitely generated module is not finitely generated in general. This will not happen if we constrain … Continue reading