Is there a term for two subsets whose intersection is {0}? In linear algebra I've just recently learned about the "direct sum", which can be defined as the sum of two vector spaces whose intersection is the null vector. 
Basically, I'm wondering if there's a formal term in abstract algebra to define algebraic substructures whose intersection is the identity element, as it seems like it'd be a, in the case of vector spaces, useful way to differentiate between subspaces that can be directly summed and subspaces that can't be.
I'm specifically asking this because algebraic substructures whose intersection is the identity element seem like they could have some interesting interacting properties that differ from the interacting properties of any two substructures (and that the term disjoint wouldn't work in the case of any algebraic substructures).
 A: I don't think there is a common term in set theory for that.
But in the context of linear algebra, it is frequent to say that two subspaces are independent when their intersection is the zero space.
A: Sets that have empty intersection are said to be disjoint. Subgroups of the same group, subspaces of the same vector space, and so on will never be disjoint, as they'll have at least one common element (the identity element of the group, the 0 vector, etc.) Sets with just one element are singletons, but it would be unusual to use that term about algebraic substructures. You can say that their intersection is trivial, or that it's the zero subgroup (in Abelian contexts), the zero subspace, etc. By a slight abuse of notation, people write this as, for example, $G \cap H = 0$, and $V \cap W = 0$.
A: In the context of subspaces of a vector space (or more generally, subgroups of a group), it is common to abuse terminology and call such subspaces disjoint.  This abuse is generally harmless since it is obviously impossible for them to actually be literally disjoint.  You could also call them (linearly) independent, though personally I think "disjoint" sounds much more natural.  I would not, however, use either term to describe arbitrary subsets (as opposed to subspaces) of a vector space whose intersection is $\{0\}$.
