We introduce the concept of almost semiprime submodules of unitary modules over a commutative ring with nonzero identity. We investigate some basic properties of almost semiprime and weakly semiprime submodules and give some characterizations of them, especially for (finitely generated faithful) multiplication modules.

Throughout this paper, all rings are commutative rings with identity and all modules are unitary. Various generalizations of prime (primary) ideals are studied in [

For any two submodules

(i) Let

(ii) Let

Let

Furthermore,

(i) It is clear that every semiprime submodule is almost semiprime. But the converse is not true in general. For example, consider

In the semiprime submodules case,

Let

Let

Let

Let

Let

(i) Let

(ii) is similar to (i).

Let

Let

The proof is straightforward.

Let

Assume that

Let

Let

if

if

Let

(ii) Let

In the following theorems, we give other characterizations of almost semiprime submodules.

Let

For

For

(i)⇒(ii) Let

(ii)⇒(iii) It is well known that if a submodule is the union of two submodules, then it is equal to one of them.

(iii)⇒(i) Let

The following theorem gives from Theorem

Let

We know that if

Let

Now in the following theorem, we give a characterization of almost semiprime submodules in (finitely generated faithful) multiplication modules. We first need the following lemma.

Let

The proof is by [

Let

(i)⇒(ii) Suppose that

(ii)⇒(i) In this direction, we need

(ii)

Let

if

It follows from [

Let

Assume that

Conversely, suppose that

For every proper ideal

Since

Let

Let

Let

(i) Let

(ii) Let

Let

Consider the

Let

Assume that

Conversely, assume that

Let

Let

Let

Let

Let

We know that every weakly semiprime is almost semiprime. Let

Now we get other characterizations of weakly semiprime submodule.

Let

For

For

(i)⇒(ii) Let

(ii)⇒(iii) It is straightforward.

(iii)⇒(i) Let

Let

Let

Let

In Theorem

Let

Assume that

Let

In [

Let

Suppose

Now we give characterizations of weakly semiprime submodules in (finitely generated faithful) multiplication modules.

Let

(i)⇒(ii) It follows from Proposition

(ii)⇒(i) In this direction, we need

(ii)

A proper ideal

Let

The proof is by [

Let

if

if

if

Note first that

(i) Assume that

The proofs of (ii) and (iii) follow from (i).

The author declares that there is no conflict of interests regarding the publication of this paper.

The research of the author is supported by a grant from Payame Noor University (PNU). The author would like to thank the referee(s) for valuable comments and suggestions which have improved the paper.