From Cayley's theorem we know that every group is a symmetric group, i.e. a group of permutations. But what happens when we "extend" a group to a ring or a field for example; is there any generalisable results for more complex algebraic structures concerning isomorphic properties just as Cayley's theorem? Or anything similar?
For me I visualize as follows: I have a commutative group $G$ say, and add the axioms needed for a ring (suppose the axioms holds so we actually get a ring). Since another binary operator now is defined it feels like the underlying symmetric group should disappear when we consider it as a ring, i.e. these permutation elements are no longer valid.
Best regards
If $M$ is a monoid with underlying set $X$, then $M \to \mathrm{End}(X)$, $m \mapsto (x \mapsto mx)$ is an injective monoid homomorphism. (If $M$ is a group, then this restricts to $M \to \mathrm{Aut}(X)=\mathrm{Sym}(X)$).
If $R$ is a ring with underlying abelian group $A$, then $R \to \mathrm{End}(A)$, $r \mapsto (a \mapsto ra)$ is an injective ring homomorphism.
If $R$ is a $k$-algebra with underlying $k$-module $V$, then the same map is an injective $k$-algebra homomorphism $R \to \mathrm{End}(V)$.
More generally, if $C$ is a closed symmetric monoidal category and $M$ is a monoid object in $C$, with underlying object $X$, then there is a monomorphism of monoids $M \to \underline{\mathrm{End}}(X)$. It is defined to be the adjoint of the multiplication morphism $X \otimes X \to X$ of $M$. Since $X \otimes X \to X$ is a split epimorphism in $C$, its adjoint is in fact a split monomorphism in $C$.
For $C=\mathsf{Set}$ we get the embedding for monoids resp. groups, for $C=\mathsf{Ab}$ we get the embedding for rings, for $C=\mathsf{Mod}(k)$ the one for $k$-algebras. There are a lot more examples for $C$. For example, $C=\mathsf{CGHaus}$ gives an embedding for (compactly generated Hausdorff) topological groups.
There is no hope to get something like that for fields. Every field embeds into an algebraically closed field, but this cannot be done canonically.