On functional prime ideals in commutative rings (Q6600727)

All rings are commutative with \(1\neq 0\) and all modules are unital. There are many generalization of the notaion of prime ideals/submodules. \textit{I. Akray} and \textit{H. M. Salih} [J. Math. Ext. 16, No. 1, Paper No. 2, 11 p. (2022; Zbl 1486.13004)], studied \(\alpha\)-prime ideal of a commutative ring: if \(R\) is a commutative ring, \(\alpha\) be a ring endomorphism of \(R\), then an ideal \(I\) of \(R\) is called an \(\alpha\)-prime ideal of \(R\), whenever \(ab\in I\), then \(a\in I\) or \(\alpha(b)\in I\), for each \(a,b\in R\).\N\NIn this paper the author studied the notation of functional prime ideal: If \(M\) is an \(R\)-module over acommutative ring, \(\phi\in \Hom_R(M,R)\), then a nonzero ideal \(I\) of \(R\) is called \(\phi\)-prime whenever \(a\phi(m)\in I\) implies that \(a\in I\) or \(\phi(m)\in I\), for each \(a\in R\) and \(m\in M\). It is shown that \(I\) is \(\phi\)-prime if and only either \(\phi(M)\subseteq I\) or \(I\) is a prime ideal of \(R\). In particular, if \(\phi\) is onto, then \(I\) is \(\phi\)-prime if and only if \(I\) is prime. Next, the author studied \(\phi\)-primes in certain integral domains such as Prufer domain, valuation domain, completely integrally closed domain, in the ring of fractions and in ring extension \(R\subseteq T\) of integral domain with \((R:T)\in \mathrm{Max}(R)\). For example, the author prove that if \(R\) is a completely integrally closed (resp. valuation) domain and \(I\) be an ideal of \(R\), then \(I\) is \(\phi\)-prime for each \(\phi\in \Hom_R(I^{-1},R)\) (resp. \(\phi\in \Hom_R(I,R)\)) if and only if \(I\) is a prime ideal.