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:For every ring containing , and every element of , there is a unique algebra homomorphism from to that fixes , and maps to .

As for all universal properties, this defines the pair up to a unique isomorphism, and can therefore be taken as a definition of .

The image of the map , that is, the subset of obtained by substituting for in elements of , is denoted and called the adjunction of a to K. For example, , and the simplification rules

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