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Ideal in a variety with zero
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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 A is an algebra in . An ideal in A is a nonempty subset S, with the following property:
For any expression 
constructed using the operators of the operator domain, such that whenever all the ujs are zero, 
takes the value zero, it is true that when all the uj are in S, takes a value inside S.
Such expressions are termed ideal terms.
Relation with other properties
Stronger properties
- Kernel of a congruence: The kernel of a congruence is defined as the inverse image of zero under the quotient map arising from the congruence. The kernel of any congruence must be an ideal. This gives a natural map from the collection of all congruences to the collection of all ideals, which need not in general be either injective or surjective. When the map is a bijection, we say that the variety is ideal-determined. The variety of groups is ideal-determined.
