On strongly homogeneous prime ideals in a graded integral domain (Q6635905)

Let \(R\) be an integral domain and \(K=Q(R)\) be its fraction field. A prime ideal \(P\) of \(R\) is called strongly prime, if whenever \(xy\in P\), with \(x,y\in K\), then \(x\in P\) or \(y\in P\). If each prime ideal of \(R\) is strongly prime, \(R\) is called a pseudo-valuation domain (PVD). If \(R=\bigoplus\limits_{\alpha\in \Gamma}R_\alpha\) is \(\Gamma\)-graded, then a homogeneous prime ideal \(P\) is called strongly homogeneous prime if whenever \(xy\in P\), with \(x,y\in R_h\), then \(x\in P\) or \(y\in P\). If every homogeneous prime is strongly homogeneous prime, then \(R\) is called a graded PVD. The aim of the paper is to study the strongly homogeneous primes and to characterize graded PVD's. Several illustrating examples are given.