Abstract Algebra: Theory and Applications · Problem 18.4.19

Lattices and Boolean Algebras — Thomas Judson

Mathematics — Abstract Algebra · descending-chain-condition

Problem

Let

D

be an integral domain with a descending chain of ideals

I_1 \supset I_2 \supset I_3 \supset \cdots

. Suppose that there exists an

N

such that

I_k = I_N

for all

k \geq N

. A ring satisfying this condition is said to satisfy the descending chain condition, or DCC. Rings satisfying the DCC are called Artinian rings, after Emil Artin. Show that if

D

satisfies the descending chain condition, it must satisfy the ascending chain condition.

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