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Let $R$ be an associative ring in which an identity element is not assumed. A right quotient ring of $R$ is an overring $S$ such that for each $a\in S$ there corresponds $r\in R$ such that $ar\in R$ and $ar\neq 0$

Is a MAXIMAL QUOTIENT RING for $R$ a ring $S$ maximal with respect to above property?

For a commutative unital ring, can we deduced that MAXIMAL QUOTIENT RING=$R_S$, where $S$ is the set of all non zero-divisors?

Is there any simple comprehensive references for maximal quotient rings?

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    $\begingroup$ The name "quotient ring" is taken by something else, how about you call this, say an "idealizer", as analogous to normalizer. $\endgroup$
    – user1688
    Commented Mar 28, 2017 at 8:28
  • $\begingroup$ May I ask what is the motivation behimd your questions? $\endgroup$
    – Xam
    Commented Mar 28, 2017 at 12:13

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