Canonical map

Examples

• If N is a normal subgroup of a group G, then there is a canonical surjective group homomorphism from G to the quotient group G/N, that sends an element g to the coset determined by g.
• If I is an ideal of a ring R, then there is a canonical surjective ring homomorphism from R onto the quotient ring R/I, that sends an element r to its coset I+r.
• If V is a vector space, then there is a canonical map from V to the second dual space of V, that sends a vector v to the linear functional f_v defined by f_v = λ.
• If is a homomorphism between commutative rings, then S can be viewed as an algebra over R. The ring homomorphism f is then called the structure map. The corresponding map on the prime spectra is also called the structure map.
• If E is a vector bundle over a topological space X, then the projection map from E to X is the structure map.
• In topology, a canonical map is a function f mapping a set X → X/R, where R is an equivalence relation on X, that takes each x in X to the equivalence class modulo R.