When graded domains are Schreier or pre-Schreier (Q1764817)

All rings and monoids are assumed to be commutative. An element \(x\) of an integral domain \(R\) is primal if whenever \(x\) divides a product \(y_1y_2\), with \(y_1,y_2\in R\), then \(x=z_1z_2\) where \(z_1\) divides \(y_1\) and \(z_2\) divides \(y_2\). An integral domain in which every element is primal is called pre-Schreier and it is called Schreier if in addition it is integrally closed. A cancellative torsion-free monoid \(M\) is called pre-Schreier if for all \(x,y_1,y_2 \in M\) such that \(x \leq y_2+y_2\), there are \(z_1,z_2 \in M\) with \(Z_1\leq y_1\) and \(z_2\leq y_2\) (we say \(a\leq b\) if \(a+c=b\) for some \(c\in M\)). A cancellative torsion-free monoid \(M\) is integrally closed if \(nx \in M\) for \(n>0\) and \(x\in G\), the quotient group of \(M\), implies that \(x\in M\). Thus an integrally closed torsion-free monoid which is pre-Schreier is called Schreier. The article investigates when an integral domain graded by a cancellative torsion-free monoid \(M\) is pre-Schreier or Schreier. They then specialize these results to a monoid ring \(A[M]\) over an integral domain \(A\). The main theorem shows that \(A[M]\) is pre-Schreier \(\Leftrightarrow\) \(A[M]\) is Schreier \(\Leftrightarrow\) \(A\) and \(M\) are Schreier.