# Syzygy

From Encyclopedia of Mathematics

An astronomical term denoting the disposition of three celestial bodies on a line.

In algebra it is used in the sense of a relationship. Let be a left -module, and let be a family of elements of ; a relationship, or syzygy, between the is a set of elements of the ring such that . Thus there arises the module of syzygies, the chain complex of syzygies, etc. See Hilbert theorem on syzygies.

#### Comments

Syzygies appear in the definition of syzygetic ideals and the theory of regular algebras and regular sequences, cf. Koszul complex; Depth of a module.

**How to Cite This Entry:**

Syzygy. V.I. Danilov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Syzygy&oldid=18722

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098