I am a beginner in this field. My question is

    (1) Is the existence of $E_\infty$ ring structure not closed under weak equivalence of ring spectra? 

    (2) If (1) is true, what is the risk of replacing a ring spectrum by a weakly equivalent $E_\infty$ ring?

I am a little confused about the construction of $BP$ in [Tilman Bauer's notes](https://math.mit.edu/events/talbot/2007/tmfproc/Chapter09/Bousfield.pdf). According to section 3 of this note, the Brown-Peterson spectrum $BP$ can be constructed by taking the quotient of $MU_p$ by an ideal $I=ker(MU_* \to BP_*)$. 

This quotient is given by theorem V.2.6. in [Rings, Modules, and Algebras in Stable Homotopy Theory](https://people.math.rochester.edu/faculty/doug/otherpapers/EKMM.pdf). By theorem V.3.2. in this book, this quotient ring $BP':=MU_p/I$ is an associated and commutative $MU$-algebras for $p>2$. Lemma VII.1.3. shows a commutative $MU$-algebras $BP'$ is a commutative $S$-algebra with a map $MU\to MP'$ of $S$-algebras. So lemma II.3.4. tells us $BP'$ is an $E_\infty$-ring which is also an $S$-algebra. In section V.4., $BP'$ is called a model of $BP$.

On the other hand, it is known that [$BP$ is not $E_\infty$ for all primes $p$](https://mathoverflow.net/questions/247526/why-do-homotopy-theorists-care-whether-or-not-bp-is-e-infty). So $BP'$ can not be exactly $BP$ in this sense.

I guess Tilman Bauer is just talking about "an $E_\infty$ model of $BP$" in his notes because this note is for a workshop of topological modular forms where we are interested in lifting commutative homotopy ring spectra to $E_\infty$ rings and the existence of $E_\infty$ ring structure is not closed under weak equivalence of ring spectra. 

If this is the case, what is the risk of doing such replacement?