… on Definition 7, we know that every prime submodule is weakly prime, and zero submodules
of … that cyclic weakly prime submodule and cyclic almost prime submodule on ℤ − module ℤ …

… prime submodule”, and in this paper we prefer to use classical prime submodule instead of
weakly prime submodule. … classical prime submodules which is called weakly classical prime …

… prime submodule is a graded weakly prime submodule, and every graded weakly prime
submodule is a graded almost prime submodule… set of all graded R-submodules of M and φ : GS…

… By setting \(S=\{1\}\), we conclude that every strongly prime submodule is an S-strongly
prime submodule by Proposition 2.2. The following example shows that the converse is not true …

… prime submodules, 2-absorbing submodules. We prove that every prime submodule is a
1-absorbing submodule, but the converse is not true: To see this, consider the cyclic submodule …

… that is R-prime submodule of S an R-prime ideal of S. We say that R ⊂ S is a prime submodule
ideal extension (for short PSI-extension) if each R-prime ideal of S is prime. We say that …

… of the prime submodules over noncommutative rings and we … that NI is almost prime, where
N is a submodule of M and I is … On the other hand, it is easy to see that in Z ⊕ Z as Z-module…

… Let p be a prime ideal of R. We say that a prime submodule N of M is p-prime if (N : M) = p.
If p is proper (ie p = R) then every p-… Our take on this is Proposition 1. …

… an S-prime ideal of R if I is an S-prime submodule of R-module R. Note that all prime
submodules P of M such that (P :R M) ∩ S = ∅ becomes an S-prime submodule since 1 ∈ S [12]. A …

… that a proper submodule N of an R-module M is a maximal non-prime submodule if N is not
a prime submodule of M but any proper submodule K of M with N ⊆ K and N ̸= K is a prime …