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Thanks for the counterexample, Jason! I also realized that my statement was not not true without assumption $pdim_{R/(f)}(S)<\infty$. I was slow to correct my claim. This is an educational counterexaple.
Thank you for the detailed answer. This helps me a lot. I now understand that whether or not the inequality holds depends not only $R$ but alos $M$. I will check the book you referred to. Thanks again!
I should have mentioned about localization. It depends on $M$. But please assume that it is possible. Examples in my mind are mostly not very noncommutative; they have plenty of normaliizng elements.
