As for an arbitrary field $K$, we know that its algebraic closure always exists and it is unique up to an isomorphism. However, when we talk about integral closure of some commutative ring $A$, we are always given $A$ as some subring of a larger ring $B$ and its closure is defined to be all the elements of $B$ integral over $A$.
It seems like, due to the lack of cancellation law for multiplication, there doesn't seem to be a natural choice for even "simple extensions" by a root of a polynomial, hence making the concept rather meaningless. It still seems, however, possible to arbitrary append $A$ with a root of a polynomial in $A[x]$ and get some "integral extension" of $A$, albeit it not being a natural choice. For example, $\overline{\mathbb{Q}}$ in $\mathbb{C}$ might be considered as an integral closure of $\mathbb{Z}$.
So here is my question: Given a commutative ring $A$, is there a ring $B$ such that $B$ is integral over $A$ and every polynomial in $A[x]$ somehow splits completely in $B[x]$?