Well-centered overrings of a commutative ring in pullbacks and trivial extensions

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 [9].