Dimension theory (algebra)

Imagine you have a set of building blocks, like Legos. Each block is different in size, shape, and color. Now, let's say we want to organize these blocks into groups based on certain characteristics. For example, we could put all the red blocks together, or all the blocks that are rectangular. This is similar to what mathematicians do with algebraic objects.

Dimension theory is a way of looking at algebraic objects and categorizing them based on certain properties. We can think of these properties as different dimensions of the object. For example, an object might have dimension 1 if it only has one distinguishing trait, or it might have dimension 2 if it has two traits that are important to consider.

One common way to think about dimension is through the concept of a basis. A basis is a set of objects that can be used to build up any other object in the system. For example, in a three-dimensional space, we might use the unit vectors (1, 0, 0), (0, 1, 0), and (0, 0, 1) as a basis for describing any point in the space.

Dimension theory helps us understand how objects behave in certain situations. For example, if we have an algebraic system that has dimension 2, we know that it's possible for there to be two different ways to get from one point in the system to another. On the other hand, if the system has dimension 1, there is only one way to move between points.

Overall, dimension theory is a way of organizing and understanding complex algebraic objects by breaking them down into simpler pieces based on their most important properties.