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I'm looking for a reference that treats basic module theory over non-associative rings, the isomorphism theorems and so on. I imagine the theory is known, but have not been able to find a reference.

Edit: Let $R$ be a non-associative ring. By a left $R$-module (similarly for a right $R$-module), I mean an abelian group (M,+) and an operation $\cdot\colon R\times M\to M$, such that for all $r_1,r_2\in R$ and all $m_1,m_2\in M$, the following hold:

  1. $(r_1+r_2)\cdot m_1 = r_1\cdot m_1 + r_2\cdot m_1$,
  2. $r_1\cdot(m_1+m_2) = r_1\cdot m_1 + r_1\cdot m_2$,
  3. $1\cdot m_1 = m_1$ if $R$ is unital.
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    $\begingroup$ You should start with a definition of what a module would be... there's a notion of module over an associative ring, over a Lie ring, but they don't match... in the first case you have the axiom $(ab)x=a(bx)$ and in the second case you have $(ab)x=a(bx)-b(ax)$. If you choose the first axiom, an $A$-module is just a module over the largest associative quotient of $A$... $\endgroup$
    – YCor
    Commented May 8, 2018 at 13:12
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    $\begingroup$ Maybe it's worth looking for something on universal algebra? $\endgroup$
    – Qfwfq
    Commented May 8, 2018 at 13:32
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    $\begingroup$ As regards the edit, you now seem to want a notion of module which is unrelated to the multiplicative structure of $R$! $\endgroup$
    – YCor
    Commented May 8, 2018 at 16:15
  • $\begingroup$ For some sensible versions see the question Do non-associative objects have a natural notion of representation? here on MO $\endgroup$
    – მამუკა ჯიბლაძე
    Commented May 8, 2018 at 16:56
  • $\begingroup$ Possible duplicate of Do non-associative objects have a natural notion of representation? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented May 8, 2018 at 16:57

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A Google search of this term brings a number of references, did you try it?

In any case, a very obvious relevant reference is the old paper of Osborn called Modules over nonassociative rings.

More recent works define modules over any kind of algebras using the language of operads, but I am not sure that this is what you want since you did not indicate your background and the research area you are coming from.

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  • $\begingroup$ Believe it or not, I am a frequent user of their service(s). I have seen that reference a couple of times before, but never been able to get hold of the paper. Will try harder, as the title and abstract really seem intriguing. $\endgroup$
    – Per
    Commented May 12, 2018 at 1:05

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