I have been reading Frobenius Algebras, Volume 1 By Andrzej Skowroński, Kunio Yamagata. On page 18 I came across the following paragraph, and I founded interesting, I will quote it and then ask my questions.
Although historically the left modules were more natural objects of study than the right modules, in the modern representation theory of algebras and from technical reasons, the right modules became more popular. We will follow this trend and concentrate on study of right modules.
When the authors say left modules are more natural objects, does that mean multiplying by the elements of the ring is motivated by applying maps? (I mean $r.m$ looks like $r(m)$, and when we consider the correspondence between representations of an algebra and it's left modules this would become more obvious)
My second question is, why do the right modules become more popular? is there any reason for that? What does "technical reasons" mean in this context?
There is no mathematical reason to prefer right or left modules. The symmetry is broken by the conventions of written English (or more generally, written European language) and mathematical exposition, in which a function applied to an element of a set is typically written $f(x)$ and function composition is $f(g(x))$; most of us would find the notation $$(x)f=((x)\mathrm{log})\mathrm{sin}$$ difficult to parse. As noted in the comments, if you start with a right module over a ring, it naturally becomes a left module over its endomorphism ring, conveniently coinciding with our conventions for writing functions applied to elements of a set.