Universal property of sum of ideals

Question

If $A$ is a commutative algebra over the field $K$, and $I,J$ are ideals of $A$ (in particular, they are also algebras over $K$), is it possible to characterize the ideal they generate together $I + J = \langle I,J \rangle$ in terms of a universal property among $K$-algebras?

Answer 1

It is the pushout of the diagram $I \leftarrow I \cap J \rightarrow J$. In other words, given a $K$-algebra $B$ and morphisms $\sigma : I \to B$ and $\eta : J\to B$ such that $\sigma|_{I\cap J} = \tau|_{I \cap J}$, there is a unique morphism $I + J \to B$ inducing $\sigma$ on $I$ and $\tau$ on $B$.