Editing Embedding (section)

==Algebra==
In general, for an [[Variety (universal algebra)|algebraic category]] <math>C</math>, an embedding between two <math>C</math>-algebraic structures <math>X</math> and <math>Y</math> is a <math>C</math>-morphism {{nowrap|<math>e:X\rightarrow Y</math>}} that is injective.

===Field theory===

In [[field theory (mathematics)|field theory]], an '''embedding''' of a [[field (mathematics)|field]] <math>E</math> in a field <math>F</math> is a [[ring homomorphism]] {{nowrap|<math>\sigma:E\rightarrow F</math>}}.

The [[Kernel (algebra)|kernel]] of <math>\sigma</math> is an [[ideal (ring theory)|ideal]] of <math>E</math>, which cannot be the whole field <math>E</math>, because of the condition {{nowrap|<math>1=\sigma(1)=1</math>}}. Furthermore, any field has as ideals only the zero ideal and the whole field itself (because if there is any non-zero field element in an ideal, it is invertible, showing the ideal is the whole field). Therefore, the kernel is <math>0</math>, so any embedding of fields is a [[monomorphism]]. Hence, <math>E</math> is [[isomorphic]] to the [[Field extension|subfield]] <math>\sigma(E)</math> of <math>F</math>. This justifies the name ''embedding'' for an arbitrary homomorphism of fields.

===Universal algebra and model theory===
{{further|Substructure (mathematics)|Elementary equivalence}}
If <math>\sigma</math> is a [[signature (logic)|signature]] and <math>A,B</math> are <math>\sigma</math>-[[structure (mathematical logic)|structures]] (also called <math>\sigma</math>-algebras in [[universal algebra]] or models in [[model theory]]), then a map <math>h:A \to B</math> is a <math>\sigma</math>-embedding exactly if all of the following hold:
* <math>h</math> is injective,
* for every <math>n</math>-ary function symbol <math>f \in\sigma</math> and <math>a_1,\ldots,a_n \in A^n,</math> we have <math>h(f^A(a_1,\ldots,a_n))=f^B(h(a_1),\ldots,h(a_n))</math>,
* for every <math>n</math>-ary relation symbol <math>R \in\sigma</math> and <math>a_1,\ldots,a_n \in A^n,</math> we have <math>A \models R(a_1,\ldots,a_n)</math> iff <math>B \models R(h(a_1),\ldots,h(a_n)).</math>

Here <math>A\models R (a_1,\ldots,a_n)</math> is a model theoretical notation equivalent to <math>(a_1,\ldots,a_n)\in R^A</math>. In model theory there is also a stronger notion of [[elementary embedding]].