I'm reading a text wich says: "Let $k\subset K$ be a skew field extension." Now I know what a field extension is, but I'm not quite sure what to think of a skew field extension. Also, how would you construct such an object?
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If $A,B$ are two arbitrary algebraic objects, then an extension $A \to B$ is by definition a monomorphism $A \to B$. This includes the notions of ring extension, field extension, skew field extension, etc. Notice that in the category of skew fields every morphism is a monomorphism. Hence, an extension of skew fields is just a homomorphism of skew fields. Often this homomorphism is assumed to be an inclusion of sets (which is not necessary and in fact, in my opinion, a little bit misleading, because basic examples such as $k \to k[x]/(f)$ are not inclusions).
Martin Brandenburg
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PS: The question "Also, how would you construct such an object?" is too general to be answered. – Martin Brandenburg Jun 09 '14 at 11:16
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2Not to spoil Martin's soup, but I have never heard of "module extension" being used for a monomorphism of modules -- instead it often means an isomorphism class of short exact sequences. I'd say the meaning of the word depends on the category, alas. – darij grinberg Jun 09 '14 at 11:52
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1That's right, thank you for that addition. – Martin Brandenburg Jun 09 '14 at 12:08
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On the other hand, the notions are compatible. An extension of modules $0 \to A \to B \to C \to 0$ is completely determined by the monomorphism $A \to B$. In fact, $B \to C$ is just a cokernel of $A \to B$. – Martin Brandenburg Jun 09 '14 at 13:47
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Ah, that's a nice viewpoint! – darij grinberg Jun 10 '14 at 00:37