Natural Projection

The natural projection, also called the homomorphism, is a logical way of mapping an algebraic structure onto its quotient structures. The natural projection pi is defined formally for groups and rings as follows.

For a group G, let N⊴G (i.e., N be a normal subgroup of G). Then pi:G->G/N is defined by pi:g|->gN. Note Ker(pi)=N (Dummit and Foote 1998, p. 84).

For a ring, let I be an ideal of a ring R. pi:R->R/I is defined be r|->r+I. Note Ker(pi)=I (Dummit and Foote 1998, p. 244).

This entry contributed by Nick Hutzler

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Dummit, D. S. and Foote, R. M. Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1998.

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Natural Projection

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Hutzler, Nick. "Natural Projection." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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