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I'm currently reading Ch. 10 in Dummit & Foote (3rd Edition) and towards the end of the first section, it defines an R-algebra. Dummit & Foote do a decent job of motivating certain definitions/ideas, but this one really seems important and just kinda comes out of nowhere. I was wondering if anyone could possibly give me some motivation or intuition to where this idea of an 'algebra' comes from? Thanks!

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  • $\begingroup$ What are you looking for in terms of motivation? The complex numbers and quaternions form algebras over the reals, and so do the octonions (though this last algebra is not associative and may not be encompassed by the definition in the textbook). $\endgroup$ – Matt Samuel Mar 8 '16 at 6:25
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    $\begingroup$ Algebras lie at the intersection of module and ring street. As such, both theories (especially when we talk about $F$-modules, that is-vector spaces) can be used as avenues of attack. $\endgroup$ – David Wheeler Mar 9 '16 at 3:29
  • $\begingroup$ To add on/clarify David's comment, basically by saying something is an $k$-algebra where $k$ is a field, you should always remember it is a ring AND a $k$-vector space. In particular, the natural place to do representation theory is over $k$ (and any extension of $k$), i.e. you want to consider $k$-algebra hom (i.e. ring hom and vector space hom) $\rho: A \to GL_n(k)$, or equivalently $A$-modules which are also $k$-vector spaces. $\endgroup$ – Aaron Mar 10 '16 at 11:47
  • $\begingroup$ Thank you all! These were just the answers that I needed. $\endgroup$ – hijasonno Mar 12 '16 at 2:01

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