A technique in algebra to make a homomorphism injective is to “mod out” the kernel.

While, to make a homomorphism surjective, one can restrict the codomain to the image.

This can be illustrated in the first isomorphism theorem (for groups) .

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Yes. In fact we use it to make Z/(x^2+1) is isomorphism to C.

ie f: x => x^2+1

key f = {x | x^2+1=0} = {x = sqrt (-1) = i}

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Oops! I meant R/(x^2+1) , not Z/(x^2+1). That was how human man-made Complex number from Real number in Renaissance 16th century Italy while trying to solve cubic equations.

There are other “man-made” examples in Math: Quarternion (1, i, j, k)

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