On Regular Modules over Commutative Rings

Characterizations of regular modules

“…By [26,Theorem2.1], Rad(N ) = N for each submodule N of M . Hence M is F-regular by [10,Theorem 2.3]. □ m ∈ M, mR satisfies any one of the statements given.…”

“…Let N be a proper submodule of M . In view of [10,Theorem 2.3], we have to prove that Rad(N ) = N . But to prove that Rad(N ) = N , by [26,Theorem 2.1], it is enough to show that m(a) = m(a 2 ) for all a ∈ R and m ∈ M .…”

“…(1)⇒( 7) Assume that M is F-regular. Then mR is an F-regular module for each m ∈ M by [8, Theorem 8.2] and [10,Proposition 2.6].…”

“…10 [18,. Theorem 7.14] If N is a pure submodule of M and M/N is finitely presented, then N is a direct summand of M .…”

On the Prime Spectrum of Torsion Modules

“…In recent decades, the theory of prime submodules has been widely considered as a generalization of the theory of prime ideals in commutative rings. There are many articles that seek to generalize the various properties of the prime ideals of a ring to the prime submodules of a module (see [5,7,9,11,12,13,15]).…”