# Ideal in a variety with zero

From Groupprops

Revision as of 15:17, 26 June 2008 by Vipul (talk | contribs) (New page: ==Definition== Let <math>\mathcal{V}</math> be a variety of algebras with zero. In other words, <math>\mathcal{V}</math> has an operator domain comprising operators with various ariti...)

## Definition

Let be a variety of algebras with zero. In other words, has an operator domain comprising operators with various arities, some universal identities satisfied by these operators, and a distinguished constant operator among these, called the *zero* operator.

Suppose is an algebra in . An **ideal** in is a subset containing zero, with the following property:

For any expression constructed using the operators of the operator domain, such that whenever all the s are zero, takes the value zero, it is true that when all the are in , takes a value inside .