Ideal in a variety with zero

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Definition

Let $V$ be a variety of algebras with zero. In other words, $V$ 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 $V$ . An ideal in $A$ is a nonempty subset $S$ , with the following property:

For any expression $φ (u_{1}, u_{2}, \dots, u_{m}, t_{1}, t_{2}, \dots, t_{n})$ constructed using the operators of the operator domain, such that whenever all the $u_{j}$ s are zero, $φ$ takes the value zero, it is true that when all the $u_{j}$ are in $S$ , $φ$ 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.

Weaker properties

Clot