Phi Classical 1-Absorbing Prime Submodule

In this paper, all rings are commutative with nonzero identity. Let M be an R-module. We introduce the concept of phi classical 1-absorbing prime submodules. A proper submodule N of M is a phi classical 1-absorbing prime submodule if whenever non units a, b, c belongs to R and m belongs to M with abcm belongs to N and does not belong to phi(N), then abm belongs to N or cm belongs to N. Many properties and characterizations of phi classical 1-absorbing prime submodules are given.
Comment: 18 pages. arXiv admin note: substantial text overlap with arXiv:2403.19659, arXiv:1507.08981, arXiv:1505.06730; text overlap with arXiv:2405.05971