Characterizing when the rings \(R\) and \(R[T]\) are integrally closed via linear equations over \(R\) (Q1305018)

Let \(R\) be a commutative ring with unit. The paper is devoted to study when \(R\) or the polynomial ring \(R[T]\) is integrally closed in terms of systems of linear equations over \(R\). Let \(ax= b\) be a proportionality, that is, a linear system with a single unknown \(x\), where \(a= (a_1,\dots, a_m)^t\) and \(b= (b_1,\dots, b_m)^t\) are column vectors with entries in \(R\). Let \(U_p(a)\) be the \(p\)-th determinantal ideal of \(a\) and \((a\mid b)\) the argumented matrix of the system. The authors show that \(R\) is an integrally closed ring iff any proportionality \(ax=b\) such that \(U_1(a)\) contains a nonzero-divisor and \(U_p(a)= U_p(a\mid b)\) for \(p=1,2\), has a solution in \(R\). They also show that \(R[T]\) is integrally closed iff \(R\) is reduced and any proportionality \(ax=b\) such that \(0:_R U_1(a)= 0\) and \(U_p(a)= U_p(a\mid b)\) for \(p=1,2\), has a solution in \(R\). Among other consequences, they prove that for a reduced ring \(R\) with \(\text{Min} (R)\) quasi-compact, \(R[T]\) is integrally closed iff \(R_M\) is an integrally closed domain for every maximal ideal \(M\) of \(R\).