  Let $R$ be a commutative ring with identity and $T(R)$ its total quotient
ring. We extend the notion of well-centered overring of an integral domain to
an arbitrary commutative ring and we investigate the transfer of this property
to different extensions of commutative rings in both integral and non-integral
cases. Namely in pullbacks and trivial extensions. Our aim is to provide new
classes of commutative rings satisfying this property and to shed light on some
open questions raised by Heinzer and Roitman in \cite{HR}.
