Is there a name for this sort of ideal?

Question

Let $k[x_1,...,x_n]$ be a polynomial ring over a field $k$ and let $R$ be an equivalence relation on $\{1,...,n\}$. Is there a name for the ideal $(x_i-x_j \mid iRj)$?

Answer 1

I have seen different terminology being used in different texts and papers for the ideal generated by elements of the form $x_i-x_j$ for $iRj$. The most common one is the "ideal of relations". Some texts also use the term "equivalence ideal".

Which of these is used depends on the context in which the said ideal came up. "Ideal of relations" is generally used in the context of algebraic varieties, where there is a need to specify the relations induced by $R$. If you want to convey the idea that $(x_i-x_j|iRj)$ represents the equations that enforce equalities between variables, then it might be advisable to use the term "equivalence ideal" or the "ideal of equalities".