## R a commutative ring and A an ideal in R with \$A=m_1···m_r=n_1···n_s\$ with \$m_i\$ distinct maximal ideals and all the \$n_j\$ distinct maximal ideals.

Let R be a commutative ring and A be an ideal in R satisfying $$A=m_1···m_r =n_1···n_s$$ with all the $$m_i$$ distinct maximal ideals and all the $$n_j$$ distinct maximal ideals. Show that $$r = s$$ and there exists a $$σ ∈ Sr$$ satisfying $$m_i = n_σ(i)$$ for all i.

I know that maximal ideal implies it being prime, and the product is contained in $$m_i$$ and $$n_j$$ for all $$i$$ and $$j$$, but I’m not sure how to proceed further.