Scalar extension Hopf algebroids

Given a Hopf algebra $H$, Brzezi\'nski and Militaru have shown that each braided commutative Yetter-Drinfeld $H$-module algebra $A$ gives rise to an associative $A$-bialgebroid structure on the smash product algebra $A \sharp H$. They also exhibited an antipode map making $A\sharp H$ the total algebra of a Lu's Hopf algebroid over $A$. However, the published proof that the antipode is an antihomomorphism covers only a special case. In this paper, a complete proof of the antihomomorphism property is exhibited. Moreover, a new generalized version of the construction is provided. Its input is a compatible pair $A$ and $A^{\mathrm{op}}$ of braided commutative Yetter-Drinfeld $H$-module algebras, and output is a symmetric Hopf algebroid $A\sharp H \cong H\sharp A^{\mathrm{op}}$ over $A$. This construction does not require that the antipode of $H$ is invertible.