A commutative domain \(D\) is called an open domain if the canonical contraction map Spec\((E)\longrightarrow\) Spec\((D), \,P \mapsto P\cap D,\,\) is an open map relative to the Zariski topology for each overring \(E\) of \(D\). A commutative ring \(A\) is called an open ring if \(A/P\) is an open domain for each \(P\in \text{Spec} (A)\). An open ring pair is a pair \((R,T)\) of commutative rings with identity elements such that \(R\) is a unital subring of \(T\) and \(S\) is an open ring for each subring \(S\) of \(T\) such that \(R\subseteq S\subseteq T\).
The authors prove that if \(R\subseteq T\) is an integral extension of commutative unital rings such that \(R\) is an open ring, \(R[a,b]\) is a going-down ring for each \(a,\,b\in T\) and \(T\) is semiquasilocal, then \((R,T)\) is an open ring pair. It is also proved that if \(R\subseteq T\) is an extension of commutative unital rings such that \(R[a,b]\) is an open ring for each \(a,\,b\in T\), then \((R,T)\) is an INC-pair.