I apologize in advance if the question is too basic.
Let $A$ be a differential graded algebra and $H^{0}A$ the 0-cohomology of $A$ (which is an ordinary ring) and $A^0$ is the 0-level of $ A$ . An element $a\in A^0$ is said to be homotopy invertible if there exists $x\in A^0$ such that $[xa-1]=[ax-1]=0\in H^0 A$.
I was wondering if there exists a weak equivalence of differential graded algebras $A\rightarrow B$ such that an element $ b\in B^0$ is strictly invertible if and only if it is homotopy invertible. Thank you in advance.
PS: The differential graded algebra A is not supposed to be bounded.