$k$-ideal of a blueprint

Let $B$ be a blueprint. A $k$-ideal of $B$ is a subset $I\subseteq B^\bullet$ such that:

1. $0\in I$;
2. $BI\subseteq I$;
3. whenever an additive relation
   $$\sum_{i=1}^{n}a_i+c\equiv\sum_{j=1}^{m}b_j$$
   holds in $B$, with every $a_i,b_j\in I$, then $c\in I$.

The third axiom is the subtractive condition. For a blueprint induced by a semiring, it specializes to the usual condition that $x,x+y\in I$ implies $y\in I$.

A proper $k$-ideal $\mathfrak p$ is prime if its complement $B^\bullet\setminus\mathfrak p$ is multiplicatively closed. These prime $k$-ideals are the points of the prime spectrum of a blueprint.