Definition talk:Algebra over Ring

Algebra vs Algebraic Structure

If I understand correctly, eventually all such uses of "algebra" will have to be renamed as "algebraic structure". Even though there is not much action here at the moment, this is the ultimate goal, right?--Julius (talk) 00:57, 21 February 2023 (UTC)

On the contrary. An "algebraic structure" is precisely defined as a set with a number of operations on it (either 1 or 2 usually, but not necessarily limited to that). Hence a semigroup, group, ring and field are all algebraic structures. An "algebra" is also precisely defined as a module (itself constructed as it is with an underlying algebraic structure with an operation on that) with yet another (bilinear) operation on that too. --prime mover (talk) 06:38, 21 February 2023 (UTC)

I see. I wonder if there is a way to generate a graphical representation of these dependencies (not for entire maths; one graph for algebras/ic structures, another graph for rings, modules, vector spaces, another one for topological, metric, normed spaces). Without any working knowledge it takes some time to realize which concepts lie on a similar level of abstraction. For example, we could use nodes for concepts and links for added structures, and the nodes at the top would correspond to the most abstract concepts (left-to-right arrangement could also work). We could first build such a graph manually, but with some standartization a change on an individual article could prompt an update on the graph. There are definitely latex packages for this purpose, but what is implentable on this site?--Julius (talk) 12:56, 21 February 2023 (UTC)

Interestingly, I've had a similar discussion with Caliburn on the subject of topological vector spaces, where it is far from clear what is the precise nature of the sets upon which the various topologies apply. I have no idea what the best approach would be here, but it's definitely an exercise worth doing. One of those things where whoever is so inspired may want to see what they can do.

We do already have something in place classifying and categorising the various breeds of algebraic structure, from semigroup to field, I think Lord_Farin might have done it, but I would need to search for it. --prime mover (talk) 13:10, 21 February 2023 (UTC)