Specifically, an open immersion factors through an isomorphism with an open subscheme and a closed immersion factors through an isomorphism with a closed subscheme.
en.wikipedia.org It is straightforward to show that multiplication is associative and commutative (up to isomorphism), and distributive over addition.
en.wikipedia.org Many common notions from mathematics (e.g. surjective, injective, free object, basis, finite representation, isomorphism) are definable purely in category theoretic terms (cf. monomorphism, epimorphism).
en.wikipedia.org A procedure called makes use of the isomorphism of every hypercube with this normal, changing the source, changes the hypercube.
en.wikipedia.org However, the isomorphism theorem gives a stronger result, because ring isomorphisms preserve multiplication while module isomorphisms (even between rings) in general do not.
en.wikipedia.org