Pure Extending Modules

• Pure extending modules are defined by requiring every pure submodule to be essential in a direct summand, refining the classical extending condition.
• They leverage purity via tensor exactness and finite system solvability, thus enabling precise decomposition and ring-theoretic characterizations.
• Applications include characterizing rings like von Neumann regular and semisimple rings through module decomposition and endomorphism-ring analysis.

Searching arXiv for the cited papers to ground the article and confirm metadata. Searching arXiv for "(Gupta et al., 31 Oct 2025)". Pure extending modules are modules in which the extending condition is imposed only on pure submodules rather than on all submodules. For a right $R$-module $M$, this means that every pure submodule $N \leq_p M$ is essential in a direct summand of $M$; equivalently, there exist submodules $D,D'$ with $M \cong D \oplus D'$ and $N \leq_e D$ (Gupta et al., 31 Oct 2025). The notion refines the classical CS, or extending, condition by incorporating purity—understood through tensor exactness, finite systems of linear equations, or tests against finitely presented modules—into the framework of essential extensions and summand decompositions (Gupta et al., 31 Oct 2025). Recent work establishes permanence properties, ring-theoretic characterizations, decomposition theorems, and links with nonsingularity, Rickart-type conditions, and morphic phenomena; it also clarifies the relation of the concept to earlier formulations and to lattice-theoretic relative extending notions (Gupta et al., 31 Oct 2025, Gupta et al., 2022, Celis-González et al., 26 Sep 2025).

1. Definition and foundational notions

Let $R$ be a ring and $M$ a right $R$-module. A submodule $M$0 is pure in $M$1, written $M$2, when any of the standard equivalent conditions holds: tensor-exactness, solvability of finite systems of linear equations in $M$3 whenever they are solvable in $M$4, or injectivity of the induced map on $M$5 for every finitely presented module $M$6 (Gupta et al., 31 Oct 2025). In the tensor formulation,

$M$7

Equivalently,

$M$8

(Gupta et al., 31 Oct 2025).

Essentiality is the classical condition

$M$9

A module is extending if every submodule is essential in a direct summand; it is pure extending if the same requirement is imposed only for pure submodules (Gupta et al., 31 Oct 2025, Gupta et al., 2022). Thus pure extending modules form a proper generalization of extending modules (Gupta et al., 2022).

The same framework introduces several associated notions. The pure closure, or purification, of a submodule $N \leq_p M$0 is denoted $N \leq_p M$1 and is the smallest pure submodule containing $N \leq_p M$2 (Gupta et al., 31 Oct 2025). A pure submodule $N \leq_p M$3 is pure-essential if every pure submodule $N \leq_p M$4 with $N \leq_p M$5 is zero. A module $N \leq_p M$6 is pure-uniform if every nonzero pure submodule of $N \leq_p M$7 is pure-essential; equivalently, any two nonzero pure submodules intersect nontrivially (Gupta et al., 31 Oct 2025).

The conceptual shift from extending to pure extending imports homological control into decomposition theory. Purity reflects exactness under tensor and compatibility with finite presentation, while essentiality and direct summands encode internal structure. This combination is central to the later decomposition theorems and ring characterizations (Gupta et al., 31 Oct 2025).

2. Permanence properties and comparison with extending modules

Pure extending modules enjoy several closure properties. Direct summands inherit the property: if $N \leq_p M$8 is pure extending and $N \leq_p M$9, then $M$0 is pure extending (Gupta et al., 31 Oct 2025). The same phenomenon was already established in the earlier treatment of the subject (Gupta et al., 2022). The mechanism is that pure submodules of a direct summand remain pure in the ambient module, and essentiality inside a summand restricts through intersection with that summand.

Finite direct sums are also stable. If $M$1, then $M$2 is pure extending if and only if both $M$3 and $M$4 are pure extending (Gupta et al., 31 Oct 2025). The 2022 account states the same equivalence in the form that $M$5 is pure extending iff each $M$6 is pure extending, relying on decomposition of pure submodules across direct sums (Gupta et al., 2022). By contrast, infinite sums need not be pure extending; a standard counterexample is $M$7 (Gupta et al., 31 Oct 2025).

Morita invariance is another structural feature. If $M$8 is an equivalence, then $M$9 is pure extending if and only if $D,D'$0 is pure extending (Gupta et al., 31 Oct 2025, Gupta et al., 2022). This depends on preservation of exactness, purity, essentiality, and direct summands under equivalences of module categories.

The relationship with classical extending modules is delicate. Every extending module is pure extending, but the converse fails (Gupta et al., 2022). Over von Neumann regular rings, however, the two notions coincide because every submodule is pure. If

$D,D'$1

then every module is flat and hence every submodule is pure; therefore $D,D'$2 is pure extending if and only if $D,D'$3 is extending (Gupta et al., 31 Oct 2025). A consequence is that finite direct sums of extending modules are extending over von Neumann regular rings (Gupta et al., 31 Oct 2025).

At the same time, submodule closure is limited. Pure submodules of a pure extending module need not be direct summands in general, and arbitrary submodules of a pure extending module need not themselves be pure extending (Gupta et al., 31 Oct 2025, Gupta et al., 2022). The 2022 paper gives a general counterexample via pure-injective hulls: a non-pure-extending module can sit as a submodule of a pure-injective, hence pure extending, module (Gupta et al., 2022). This marks a significant distinction from stronger decomposition classes.

3. Sufficient conditions and ring-theoretic characterizations

Several broad classes of modules are pure extending. A module $D,D'$4 is pure extending if it is fully invariant in its pure-injective envelope, quasi-pure-injective, pure-split, or flat and cotorsion (Gupta et al., 31 Oct 2025). The earlier paper also records that pure-injective modules and pure quasi-injective modules are pure extending, and that flat cotorsion modules are pure extending via quasi pure-injectivity (Gupta et al., 2022). For $D,D'$5-modules, finitely generated modules and divisible modules are pure extending (Gupta et al., 31 Oct 2025); similarly, every finitely generated module over a Noetherian ring is pure extending because its pure submodules split (Gupta et al., 2022).

Ring classes admit precise characterizations through pure extending modules. One such result states that $D,D'$6 is von Neumann regular if and only if every pure extending right $D,D'$7-module is flat (Gupta et al., 31 Oct 2025, Gupta et al., 2022). The proof uses the pure exact sequence

$D,D'$8

and the fact that pure-injective envelopes are pure extending (Gupta et al., 31 Oct 2025).

Another major characterization concerns semisimplicity. For a ring $D,D'$9, the following are equivalent: $M \cong D \oplus D'$0 is semisimple; every pure $M \cong D \oplus D'$1 module is projective; every pure $M \cong D \oplus D'$2 module is projective; every quasi-pure-injective module is projective; every pure-injective module is projective; and every pure extending module is projective (Gupta et al., 31 Oct 2025). This theorem appears in essentially the same form in the 2022 paper (Gupta et al., 2022). The significance is that pure extending modules sit inside a hierarchy of purity-sensitive analogues of classical summand conditions, and projectivity of the whole class forces semisimplicity.

Local and PDS rings provide further extremal cases. If $M \cong D \oplus D'$3 is local, then $M \cong D \oplus D'$4 is pure extending and its only pure submodules are $M \cong D \oplus D'$5 and $M \cong D \oplus D'$6 (Gupta et al., 31 Oct 2025, Gupta et al., 2022). If $M \cong D \oplus D'$7 is a PDS ring, so that every pure submodule of every module splits, then every $M \cong D \oplus D'$8-module is pure extending (Gupta et al., 31 Oct 2025, Gupta et al., 2022). These examples show that pure extending ranges from a mild condition in some settings to a universal one in others.

A related enlargement is RD-pure extending. RD-purity imposes only the elementwise conditions $M \cong D \oplus D'$9 for all $N \leq_e D$0; every pure submodule is RD-pure, but not conversely (Gupta et al., 31 Oct 2025). The 2025 paper states that the class of RD-pure extending modules properly contains pure extending modules (Gupta et al., 31 Oct 2025), while the 2022 paper phrases the relation differently, asserting that every RD-pure extending module is pure extending and that the two notions coincide on flat modules (Gupta et al., 2022). This suggests a terminological or convention-sensitive subtlety in the literature around the relative strength of the two conditions. What is unambiguous is the flat case: for projective modules, and more generally flat modules, RD-purity coincides with purity (Gupta et al., 31 Oct 2025, Gupta et al., 2022).

4. Decomposition theory and the generalized Osofsky–Smith theorem

A central development is the extension of classical uniform decomposition results to the pure setting. The paper "On Modules Whose Pure Submodules Are Essential in Direct Summands" proves a pure analogue of the Osofsky–Smith theorem: if $N \leq_e D$1 is a cyclic right $N \leq_e D$2-module such that every proper cyclic factor module of $N \leq_e D$3 is pure extending, then

$N \leq_e D$4

(Gupta et al., 31 Oct 2025).

The proof combines several structural ingredients. First, an indecomposable pure extending module is pure-uniform: if $N \leq_e D$5 is nonzero and essential in a direct summand $N \leq_e D$6, then indecomposability forces $N \leq_e D$7, so $N \leq_e D$8 is pure-essential in $N \leq_e D$9 (Gupta et al., 31 Oct 2025). Second, from the hypothesis on cyclic factors, one derives that every cyclic factor is artinian. The argument passes to descending chains, replaces the terms by their purifications $R$0, and uses stabilization of uniform dimension among summands in which these purifications are essential (Gupta et al., 31 Oct 2025). Third, if every factor of a cyclic module is endoartinian, then the module itself is endoartinian; endoartinian modules then decompose into finitely many indecomposable summands, each of which is pure-uniform in the pure extending setting (Gupta et al., 31 Oct 2025).

This theorem is formally parallel to the classical Osofsky–Smith result, where one concludes finite direct sums of uniform modules under extending-type hypotheses on cyclic factors. Over von Neumann regular rings, where pure extending and extending coincide, the pure theorem recovers a direct sum of uniform submodules and therefore reproduces a version of the classical conclusion (Gupta et al., 31 Oct 2025).

The importance of this result lies in showing that restricting the extending condition to pure submodules still preserves significant decomposition power. The passage from uniform to pure-uniform reflects the replacement of arbitrary submodules by pure ones throughout the theory, but the finite direct-sum structure survives.

5. Nonsingularity, Noetherian hypotheses, and endomorphism-theoretic consequences

Pure extending modules interact strongly with nonsingularity in finitely generated Noetherian contexts. For a module $R$1, the singular submodule is

$R$2

A module is nonsingular when $R$3 (Gupta et al., 31 Oct 2025).

Under these hypotheses, pure extending yields a Rickart-type regularity property. If $R$4 is right Noetherian and $R$5 is a finitely generated, nonsingular, pure extending right $R$6-module, then $R$7 is $R$8-Rickart (Gupta et al., 31 Oct 2025). Here Rickart means that for every endomorphism $R$9, the kernel has the form $M$0 for some idempotent $M$1 in $M$2; $M$3-Rickart means that all direct sums $M$4 are Rickart (Gupta et al., 31 Oct 2025).

The proof uses purification of kernels in arbitrary direct sums. For $M$5 and $M$6, one sets $M$7 and $M$8. Because $M$9 is Noetherian and pure extending, $R$0 inherits purity and pure extending, so $R$1 is essential in a summand $R$2 (Gupta et al., 31 Oct 2025). If $R$3, then local finite presentability and nonsingularity of $R$4 yield a contradiction via a nonzero finitely presented singular submodule. Hence $R$5 is pure and essential in $R$6; Noetherianity then forces $R$7, so kernels are direct summands (Gupta et al., 31 Oct 2025).

These arguments link purity and nonsingularity to endomorphism-ring regularity. The paper further notes that in centrally quasi-morphic modules, $R$8-Rickart and $R$9-d-Rickart coincide, and under finite uniform dimension, $M$00-Rickart together with $M$01-d-Rickart is equivalent to strong $M$02-endoregularity (Gupta et al., 31 Oct 2025). The latter is expressed by

$M$03

This places pure extending modules within a larger endomorphism-theoretic framework.

A plausible implication is that pure extending modules serve as a structural intermediary between homological purity conditions and strong internal regularity conditions on endomorphism rings. That interpretation is directly supported by the way purity, essentiality, and kernel-splitting interact in the Noetherian nonsingular regime (Gupta et al., 31 Oct 2025).

6. Examples, counterexamples, and lattice-theoretic interpretation

Examples separate pure extending sharply from classical extending. Over $M$04, the module $M$05 is pure extending because over the Noetherian ring $M$06 every pure submodule of a finitely generated module splits, but it is not extending: the submodule generated by $M$07 is not essential in any summand (Gupta et al., 31 Oct 2025). The 2022 paper records the analogous example $M$08 (Gupta et al., 2022). There is also an upper triangular matrix example producing a module that is pure extending but not extending (Gupta et al., 31 Oct 2025, Gupta et al., 2022).

Nonexamples are equally important. The product $M$09 is not pure extending because the direct sum $M$10 is pure in it but not essential in any direct summand (Gupta et al., 2022). Infinite direct sums may fail as well; $M$11 is not pure extending (Gupta et al., 31 Oct 2025). For RD-pure extending, the module $M$12 distinguishes the RD-pure and pure theories (Gupta et al., 31 Oct 2025).

The lattice-theoretic reformulation clarifies the underlying geometry. The submodule lattice $M$13 is bounded, complete, and modular, with meet given by intersection and join by sum (Celis-González et al., 26 Sep 2025). When $M$14 is instantiated as the class of pure submodules, pure-extending means precisely that $M$15 is weakly type-2 $M$16-extending: for each pure submodule $M$17, there exists a closed element $M$18 with $M$19 and $M$20 a direct summand element (Celis-González et al., 26 Sep 2025). Under idiom hypotheses—completeness, modularity, and upper continuity—this implies a weak type-1 formulation in terms of pseudo-complements (Celis-González et al., 26 Sep 2025).

This lattice perspective does not change the module-theoretic definition, but it places pure extending modules within a general theory of extending lattices relative to a distinguished class $M$21 (Celis-González et al., 26 Sep 2025). It also isolates the role of essential closures and pseudo-complements in controlling decompositions of pure submodules. This suggests that pure extending modules are one instance of a broader relative extending paradigm rather than an isolated module-theoretic curiosity.

In summary, pure extending modules preserve much of the decomposition-theoretic content of extending modules while replacing the ambient category of all submodules by the homologically significant class of pure submodules. Their theory is Morita invariant, stable under finite direct sums, aligned with classical extending over von Neumann regular rings, and rich enough to support uniform-type decomposition theorems, Rickart consequences, and counterexamples distinguishing centrally quasi-morphic from centrally morphic behavior (Gupta et al., 31 Oct 2025).