Prime and Maximal Ideals

4 min read

Consider the natural numbers $\mathbb{N}$ and order them by divisibility: $a \leq b$ whenever $b$ divides $a$. For example if I give you the set of numbers $${6,12,18,24,30}$$ Then the "largest" number in terms of divisibility is $6$.

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Noetherian modules and a short exact sequence for quotients

3 min read

Equivalent characterizations of the Noetherian condition are plentiful (see Hilbert's Basis Theorem for more) and using them interchangably can be a convenient and succinct way to express proofs. In this post, we explore yet another characterization of Noetherian conditions and bring attention to an important short exact sequence related to quotient constructions which helps clarify why this condition should hold.

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