In this section we introduce and study *α*-almost Artinian modules. We extend some of the basic results of almost Artinian modules to *α*-almost Artinian modules.

We begin with our definition of *α*-almost Artinian modules.

*An R-module* *M* *is called α-almost Artinian, if for each nonzero submodule* *N* *of* *M*, $k-\mathrm{dim}\frac{M}{N}<\alpha $ *and* *α* *is the least ordinal number with this property*.

We should remind the reader that the above concept is in fact the dual of *α*-almost Noetherian modules, see [15, Definition 1.6]. Clearly each *α*-almost Artinian module *M*, where *α* ∊ {0,1}, is almost Artinian (note, in fact if *α* = 0 then *M* is simple, i.e., it is 0-critical and if *α* = 1, then it is either Artinian or 1-critical). We thus consider the condition of a module being *α*-almost Artinian as a generalization of the condition of a module being almost Artinian.

*If* *M* *is an α-almost Artinian module, then* *M* *has Krull dimension and* *k*-dim *M* ≤ *α*. *In particular*, *k*-dim *M* = *α* *if and only if* *M* *is α-critical*.

*If* *M* *is a module with* *k*-dim *M* = *α*, *then either* *M* *is α-critical, in which case it is α-almost Artinian, or it is* *α* + *1-almost Artinian*.

*If* *M* *is an α-almost Artinian module, then either* *M* *is α-critical or* α = *k*-dim *M* + 1. *In particular, if* *M* *is an α-almost Artinian module, where* *α* *is a limit ordinal, then* *M* *is α-critical*.

The following is now immediate.

*Let* *M* *be a* *β* + *1-almost Artinian module, then either* *k*-dim *M* = *β* *or* *k*-dim *M* = *β* + 1.

*An R-module* *M* *has Krull dimension if and only if* *M* *is α-almost Artinian for some ordinal α*.

*Every α-almost Artinian module has finite uniform dimension*.

By Corollary 2.8, every *α*-almost Artinian module admits finite indecomposable direct decompositions. The next proposition provides criteria for an *α*-almost Artinian module to be indecomposable.

*Let* *M* *be an α-almost Artinian module. Then* *M* *is indecomposable if either* *α* *is a limit ordinal or* *k*-dim *M* = *α*.

The following corollary is now immediate.

*If* *M* *is an α-almost Artinian module, then either* *M* *is indecomposable or* *k*-dim *M* = *β*, *where* *α* = *β* + 1.

The following lemma which is the dual of [10, Proposition 2.2] and the next few results are needed for our study in this article.

*If* *R* *is a commutative ring and* *M* *is an α-critical module, then for each* *r* ∊ *R* *we have either Ann*_{M}(*r*) = 0 *or Ann*_{M}(*r*) = *M*.

The following result is now immediate.

*Let* R *be a commutative ring and* *M* *be an α-critical R-module, then Ann*_{R}(*M*) *is a prime ideal of R*.

The following corollary, being a trivial consequence of the previous fact, is a generalization of [14, Theorem 1.1, c].

*Let* *R* *be a commutative ring and* *M* *be an α-almost Artinian module. If* *α* ≠ *k*-dim *M* + 1, *then* *Ann*_{R}(*M*) *is a prime ideal of R*.

*Let R be a commutative ring and M be an α-critical R-module, then M is a torsion-free α-critical* $\frac{R}{An{n}_{R}(M)}$-module.

In view of the previous lemma and Lemma 2.5, the following corollary is now immediate.

*Let* *R* *be a commutative ring and* *M* *be an α-almost Artinian module. If* *α* ≠ *k*-dim *M* + 1, *then* *M* *is a torsion-free α-critical* $\frac{R}{An{n}_{R}(M)}$-module.

We also have the following lemma about critical modules.

*Let R be a commutative ring. If* *M* *is an α-critical R-module, then* *M* *is isomorphic to a submodule of the quotient field of* $\frac{R}{An{n}_{R}(M)}$.

In view of the previous lemma and Lemma 2.5, the following corollary, which is a generalisation of [14, Theorem 1.1, f], is now immediate.

*Let* R *be a commutative ring and* *M* *be an α-almost Artinian module. If* α ≠ *k*-dim *M* + 1, *then* *M* *is isomorphic to a submodule of the quotient field of* $\frac{R}{An{n}_{R}(M)}$.

In the case of critical modules we have the following proposition.

*If* *M* *is an α-critical R-module, then* End_{R} *(M)* *has no nonzero zero divisors*.

In view of the previous proposition and Lemma 2.5, we have the following immediate corollary which is a generalization of [14, Theorem 1.1, g].

*Let* *M* *be an α-almost Artinian module. If α* ≠ *k*-dim *M* + 1, *then* End_{R}*(M)* *has no nonzero zero divisors*.

A commutative ring R is called *α*-almost Artinian ring, for some ordinal number a, if for every non-zero ideal *I* of R, $k-\mathrm{d}\mathrm{i}\mathrm{m}\frac{R}{I}<\alpha $ and α is the least ordinal number with this property. We now have the following theorem.

*Let* *R be a commutative ring. If R is β+1-almost Artinian, then either* *k*-dim *R* = β *or R is a β+1-critical domain*.

The next theorem is a generalization of [14, Theorem 1.3].

*Let M be an α-critical R-module, where R is a commutative ring. Then there exists a prime ideal P such that $\frac{R}{P}$ is an α-critical domain. In particular, if M contains a torsion-free element (i.e., there exists x ∊ M such that ann.(x) = 0), then R itself is an α-critical domain*.

The next result is dual of [10, Corollary 2.4].

*If* *M* *is an R-module, then the following are equivalent*:

Next, we show that there is a long chain of submodules of a Noetherian module *M*, whose factor modules are critical of the same dimension α, where α is any ordinal less than *k*-dim *M*.

*If* *M* *is a Noetherian module which is not Artinian, that is* *k*-dim *M* ≥ 1. *Then for each ordinal* α < *k*-dim *M* *there exists an infinite chain of submodules* *M* ⊃ *M*_{1} ⊃ *M*_{2} ⊃ … *in* *M* *such that* $\frac{{M}_{i}}{{{M}_{i}}_{+1}}$ *is α-critical for each i*.

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