# Tag Archives: flatness

## Pure exact sequences

Over the next few posts, I’ll talk more about axiomatizability of algebraic structures in first-order logic. Before I do that, we need to know about purity of exact sequences. So let’s fix a ring $R$. An exact sequence $$0\to A\to B\to C\to 0$$ in the category of left $R$ modules is called pure if for […]

## Basic Examples of the Tensor Product and Flatness

The tensor product is one of the most important constructions in mathematics, and here we shall see my favourite examples of the tensor product in action, hopefully to illuminate its properties for beginners. Proofs or references are provided, but since the emphasis is on examples, the proofs that are given are terse and details are […]