In this article, the authors show the following: Theorem 1.3. Let \(A\) be a Noetherian ring with canonical module \(\omega_ A\), and suppose that \(A\) is generically a Gorenstein ring. Then \(B=\text{Hom}_ A(\omega_ A,\omega_ A)\) is the minimal extension of \(A\) with the property \((S_ 2)\). Theorem 1.6. Let \(A\) be a Noetherian integral domain and let \(B\) be a finite extension of \(A\) with the same field of fractions. Then \(A=B\) if and only if \(\dim_ AB<\infty\). Theorem 2.2. Let \(R\) be a Noetherian ring with \((S_{k+1})\) and let \(I\) be an ideal of \(R\) such that height \(I\geq k+1\). Then \(R[It]\) has \((S_{k+1})\) if and only if \(gr_ I(R)\) has \((S_ k)\). Theorem 2.5. Let \(R\) be a Cohen-Macaulay ring and let \(I\) be an equimultiple ideal of codimension \(g\geq 1\). If \(R[It]\) has the \((S_ 2)\) condition then all the powers \(I^ n\) are unmixed ideals. Theorem 2.6. Let \((R,{\mathfrak m})\) be a Gorenstein local ring of Krull dimension \(d\) and let \(I\) be an \({\mathfrak m}\)-primary, perfect, Gorenstein ideal. If the associated graded ring \(gr_ I(R)\) satisfies the condition \((S_{d-2})\), then \(I\) is a complete intersection.