overring

Let $R$ be a commutative ring having regular elements and let $T$ be the total ring of fractions of $R$.  Then  $R\subseteq T$.  Every subring of $T$ containing $R$ is an overring of $R$.

Example.  Let $p$ be a rational prime number.  The $p$-integral rational numbers (http://planetmath.org/PAdicValuation) are the quotients of two integers such that the divisor (http://planetmath.org/Division) is not divisible by $p$.  The set of all $p$-integral rationals is an overring of $\mathbb{Z}$.