Linear equations over commutative rings and grade theory (Q1188168)

Let \(R\) be a commutative ring with unity. A system of linear equations \((*)\;Ax^ T=b^ T\) is considered, where \(A=(a_{ij})\) is a \((m\times n)\)-matrix, and \(b=(b_ 1,\dots,b_ m)\), both with entries in \(R\), and \(x=(x_ 1,\dots,x_ n)\) is the unknown vector. For each integer \(p\geq 0\), \(U_ p(A)\) denotes the ideal of \(R\) generated by all \((p\times p)\)- minors of \(A\). Setting \(U_ 0(A)=R\) and \(U_ p(A)=\{0\}\) for \(p\geq\min(n,m)\), we have \((**)\;R=U_ 0(A)\supset U_ 1(A)\supset\cdots\supset\{0\}\). The rank of \(A\) is the largest \(p\) such that \(U_ p(A)\neq\{0\}\). Sufficient conditions for the solvability of the linear equations \((*)\) in terms of the rank of \(A\) and the polynomial grade of the ideals \((**)\) are given.