Definition:Canonical Injection (Abstract Algebra)

Theorem
Let $$\left({S_1, \circ_1}\right)$$ and $$\left({S_2, \circ_2}\right)$$ be algebraic structures with identities $$e_1, e_2$$ respectively.

Then the following mappings:


 * $$in_1: \left({S_1, \circ_1}\right) \to \left({S_1, \circ_1}\right) \times \left({S_2, \circ_2}\right): \forall x \in S_1: in_1 \left({x}\right) = \left({x, e_2}\right)$$


 * $$in_2: \left({S_2, \circ_2}\right) \to \left({S_1, \circ_1}\right) \times \left({S_2, \circ_2}\right): \forall x \in S_2: in_2 \left({x}\right) = \left({e_1, x}\right)$$

are monomorphisms.

These are called the canonical injections.