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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?

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    $\begingroup$ Left modules were more common, because, traditionally, scalars were written to the left of vectors in vector spaces. Right modules are handier, because they naturally become left modules over their endomorphism ring, without any need to do acrobatics with the opposite ring. $\endgroup$
    – egreg
    Jun 20, 2015 at 16:03
  • $\begingroup$ @egreg do you mean only (or mostly) when one studies the End of a module consider right modules? to play with a handier object later. $\endgroup$
    – Math137
    Jun 20, 2015 at 16:09
  • $\begingroup$ A common device in module theory is to write morphisms on the opposite side to the scalars, so most problem with endomorphisms disappear. $\endgroup$
    – egreg
    Jun 20, 2015 at 16:12
  • $\begingroup$ I imagine the reason for writing morphisms $f$ on the opposite side from scalars $\alpha$ is to make the requirement (in the definition of "morphism") that morphisms respect scalar multiplication look like an instance of associativity $f(x\alpha)=(f(x))\alpha$. Then one can just omit parentheses and write $fx\alpha$. $\endgroup$ Jun 20, 2015 at 16:26
  • $\begingroup$ @AndreasBlass That's also a reason. $\endgroup$
    – egreg
    Jun 20, 2015 at 16:27

1 Answer 1

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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.

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  • $\begingroup$ I like your argument about the convention of written english, I have seen that in arabic (when they use arabic letters in equations) they write morphis form right. $\endgroup$
    – Math137
    Jun 20, 2015 at 23:32

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