# Collective name for algebraic structures

I am doing a thesis about various algebraic structures, primarely about groups, rings and modules (with maybe hint of algebras). However always having type out ALL of them constantly gets very tedious very quickly and it's also annoying reading. So my question is, is there any collective name for all 3 (or 4)?

And with that also is there any collective name for normal subgroup, ideal and submodule? That is the structure that makes quotient structures possible.

If so are there any references for it?

• I doubt there is anything better than "algebraic structure". Concerning quotients - these are special cases of congruence from universal algebra, though it takes some work to connect the definitions. See en.wikipedia.org/wiki/Universal_algebra – lisyarus Feb 12 '16 at 13:50
• Algebraic structure to some degree almost feel like it could get even bigger which is why I wonder if there is something shorter more specific – Zelos Malum Feb 12 '16 at 13:51
• Normal subgroups and submodules are examples of normal subobjects (in the sense of category theory). Ideals of rings don't fit into the category-theoretic picture so well. I suppose you could call all of them "kernels". – Zhen Lin Feb 12 '16 at 13:55
• Could just call them "objects" if you only use the term in that way, and make it clear at the outset that you intend to do so. – MPW Feb 12 '16 at 14:37

In algebraic geometry, there is a notion of a presheaf of abelian groups on a topological space $X$. To every open set $U$ of $X$, there is associated an abelian group $\mathcal F(U)$ such that a bunch of properties hold.

But of course one is interested also in sheaves of rings, and sheaves of algebras over a ring. Since so many of the results can be stated in generality, many authors just refer to a sheaf of "sets with structure." So maybe those are the words you're looking for?

I am not sure "sets with structure" has a very precise definition though. It is usually understood as "groups, topological spaces, rings, algebras etc." Maybe one could consider a category $\mathscr C$ together with a functor $F$ into $\textrm{Set}$, and define a set with structure to be a pair $(A, F(A))$ where $A$ is an object of $\mathscr C$?

• could you point to a source where "sheaf" is used in that respect? – Zelos Malum Feb 13 '16 at 16:07
• Regarding your last paragraph, look up concrete category. But also look a the examples and counterexamples... – Najib Idrissi Feb 13 '16 at 16:09
• @zelos I've seen it a lot, but one specific example I recall is Qing Liu's book. – D_S Feb 13 '16 at 17:07
• what's it called? – Zelos Malum Feb 13 '16 at 17:11
• Algebraic Geometry and Arithmetic Curves – D_S Feb 13 '16 at 17:41

Unfortunately there are no collective name for this three structures (stylistic you could refer to them as 'this structures' or 'this three structures') and neither for the similar ideas of normal subgroups, ideals and submodules.

What they have in common is that they are magmas or combination of magmas, certain axioms applied to sets, but that won't help you.

The similarities of the subgroups-ideals-submodules are somewhat illusive. Ideals and submodules are normal subgroups because of the commutativity, but the closeness of the action of the ring have no counterpart in the case of normal groups.

If you include all mathematical structures applied to sets (e.g. even topological spaces) you could call the structures 'constructs' or 'concrete categories', but then you might have linguistic confusion while communicating with category theorists.