Abstract algebra (mathematics)
Abstract algebra is a branch of mathematics which focuses on the study of algebraic structures. It mainly studies the abstract aspect of algebra. In other words, abstract algebra doesn't care about what is exactly in an algebraic structure, but cares if this algebraic structure is equivalent to another algebraic structure.
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Proof of the isomorphism $\textrm{Hom}_{A-mod}(\textrm{Hom}_{A-mod}(A,A),A)\cong A$ for any ring $A$
Let $A$ be a commutative ring with identity. Then we have an isomorphism $$A \rightarrow \textrm{Hom}_{A-mod}(A,A), a \mapsto (x \mapsto ax)$$, whose inverse is $$\textrm{Hom}_{A-mod}(A,A) \rightarrow A, \varphi \mapsto \varphi(1).$$Then applying this isomorphism, we get $$\textrm{Hom}_{A-mod}( \textrm{Hom}_{A-mod}(A,A) ,A) \cong \textrm{Hom}_{A-mod}(A,A) \cong A.$$
2024-06-04 01:08:24