u-S-Semisimple Modules in Homological Algebra

• u-S-semisimple modules are defined so that every u-S-exact sequence with the module in the middle splits after multiplication by an element of S.
• They establish a uniform homological framework that links semisimplicity, projectivity, and injectivity in module and ring theory.
• Examples show that while classically semisimple modules are u-S-semisimple, non-semisimple rings can also exhibit u-S-semisimplicity under specific multiplicative subsets.

Searching arXiv for recent and foundational papers on u-S-semisimple modules and related uniform-S homological notions. u-$S$-semisimple modules are a uniform-$S$ generalization of semisimple modules in which exactness and splitting are weakened by allowing failure to be annihilated by a single element of a fixed multiplicative subset $S\subseteq R$. For a commutative ring $R$ with identity and a multiplicative subset $S$, an $R$-module $M$ is called u-$S$-semisimple if every short u-$S$-exact sequence

$0\to A\to M\to C\to 0$

is u-$S$0-split; equivalently, every extension with middle term $S$1 splits after multiplying the relevant identity morphism by some element of $S$2 (Zhang et al., 2021). This notion sits inside the broader uniform-$S$3 homological framework, where kernels, cokernels, Ext-groups, and splitting conditions are controlled uniformly by one element of $S$4, and it provides an $S$5-relative analogue of the classical equivalences among semisimplicity, projectivity, injectivity, and splitting of short exact sequences (Zhang et al., 2021, Zhang et al., 14 Feb 2026).

1. Uniform-$S$6 framework and the definition

Let $S$7 be a commutative ring with identity and $S$8 a multiplicative subset. The basic uniform notions are formulated by replacing exactness and vanishing with annihilation by a single element of $S$9. An $S\subseteq R$0-module $S\subseteq R$1 is u-$S\subseteq R$2-torsion if there exists $S\subseteq R$3 such that $S\subseteq R$4. A homomorphism $S\subseteq R$5 is a u-$S\subseteq R$6-monomorphism if $S\subseteq R$7 is u-$S\subseteq R$8-torsion, a u-$S\subseteq R$9-epimorphism if $R$0 is u-$R$1-torsion, and a u-$R$2-isomorphism if both conditions hold (Zhang et al., 2021).

A sequence

$R$3

is u-$R$4-exact if there exists $R$5 such that

$R$6

Accordingly, a short sequence

$R$7

is a short u-$R$8-exact sequence when it is u-$R$9-exact and $S$0, $S$1 are respectively a u-$S$2-monomorphism and a u-$S$3-epimorphism (Zhang et al., 2021, Zhang et al., 14 Feb 2026).

The splitting condition is likewise weakened. A short u-$S$4-exact sequence

$S$5

is u-$S$6-split if there exist $S$7 and a map $S$8 such that

$S$9

Equivalently, there exist $R$0 and $R$1 such that

$R$2

(Zhang et al., 2021). The module-level notion is then defined by internalizing this condition at the middle term:

An $R$3-module $R$4 is u-$R$5-semisimple if every short u-$R$6-exact sequence $R$7 is u-$R$8-split (Zhang et al., 2021).

This definition is the precise uniform-$R$9 analogue of the classical characterization of semisimple modules by splitting of all short exact sequences with middle term $M$0. The change is not merely formal: the defect from ordinary splitting is uniformly controlled by a single element of $M$1, which is stronger than allowing different annihilators for different elements or submodules (Zhang et al., 14 Feb 2026).

2. Equivalent formulations and immediate consequences

A central simplification is that the definition may be tested on ordinary short exact sequences, not only on those already known to be u-$M$2-exact. Specifically, $M$3 is u-$M$4-semisimple if and only if every short exact sequence

$M$5

is u-$M$6-split (Zhang et al., 2021). This equivalence is conceptually important: u-$M$7-semisimplicity is not just a property internal to the weakened exact structure, but a robust statement about all extensions with middle term $M$8.

Several immediate consequences follow.

First, every classically semisimple module is u-$M$9-semisimple, since ordinary splitting corresponds to the special case $S$0 (Zhang et al., 2021). Second, every u-$S$1-torsion module is u-$S$2-semisimple, because multiplication by a suitable $S$3 annihilates the relevant maps (Zhang et al., 2021). These two facts show that u-$S$4-semisimplicity interpolates between genuine semisimplicity and modules whose deviation from semisimplicity is uniformly killed by $S$5.

A further structural property is permanence along short u-$S$6-exact sequences: if

$S$7

is u-$S$8-short exact and $S$9 is u-$S$0-semisimple, then both $S$1 and $S$2 are u-$S$3-semisimple (Zhang et al., 2021). This mirrors the classical behavior of semisimple objects under split exact sequences, although here the mechanism is mediated by uniform $S$4-splitting rather than genuine decomposition.

It is also important not to conflate the module notion with the ring notion. A ring $S$5 may be u-$S$6-semisimple as a module over itself without being a u-$S$7-semisimple ring. The case $S$8, $S$9, exhibits exactly this phenomenon: $0\to A\to M\to C\to 0$0 is u-$0\to A\to M\to C\to 0$1-semisimple as a $0\to A\to M\to C\to 0$2-module, but $0\to A\to M\to C\to 0$3 is not a u-$0\to A\to M\to C\to 0$4-semisimple ring (Zhang et al., 2021).

3. Homological characterization and relation to u-$0\to A\to M\to C\to 0$5-projectivity

The theory becomes more transparent when viewed homologically. In the same framework, an $0\to A\to M\to C\to 0$6-module $0\to A\to M\to C\to 0$7 is u-$0\to A\to M\to C\to 0$8-projective if for every short u-$0\to A\to M\to C\to 0$9-exact sequence

$S$00

the induced sequence

$S$01

is u-$S$02-exact (Zhang et al., 2021). This condition admits several equivalent formulations, notably that $S$03 is u-$S$04-torsion for every $S$05-module $S$06, and in fact that $S$07 is u-$S$08-torsion for all $S$09 and all $S$10 (Zhang et al., 2021).

The relation to u-$S$11-semisimplicity at the module level is not simply an equivalence in arbitrary rings, but the two notions become tightly linked in ring-theoretic contexts where all modules share the same uniform homological behavior. In particular, over a u-$S$12-semisimple ring, every module is simultaneously u-$S$13-semisimple, u-$S$14-projective, and u-$S$15-injective (Zhang et al., 2021, Zhang et al., 14 Feb 2026).

More refined characterizations are available through relative notions. An $S$16-module $S$17 is u-$S$18-projective relative to $S$19 if for any u-$S$20-epimorphism $S$21, the induced map

$S$22

is a u-$S$23-epimorphism; dually, relative u-$S$24-injectivity is defined via u-$S$25-monomorphisms into $S$26 (Adarbeh et al., 1 Sep 2025). With these definitions, an $S$27-module $S$28 is u-$S$29-semisimple if and only if every $S$30-module is u-$S$31-injective relative to $S$32, and equivalently if and only if every $S$33-module is u-$S$34-projective relative to $S$35 (Adarbeh et al., 1 Sep 2025). This gives a precise homological interpretation: a u-$S$36-semisimple module is one that makes the entire module category relatively projective and injective against it.

The same paper also shows that u-$S$37-semisimple modules are necessarily u-$S$38-quasi-projective and u-$S$39-quasi-injective (Adarbeh et al., 1 Sep 2025). This generalizes the familiar classical implication that semisimple modules are both quasi-projective and quasi-injective, but again only in the weakened, uniform-$S$40 sense.

4. Ring-level theory and collapse of the relative homological structure

The ring-level notion is defined by requiring free modules to be u-$S$41-semisimple. Thus $S$42 is a u-$S$43-semisimple ring if every free $S$44-module is u-$S$45-semisimple (Zhang et al., 2021, Zhang et al., 14 Feb 2026). The central theorem gives a full homological collapse analogous to the classical semisimple-ring situation:

For a ring $S$46 and multiplicative subset $S$47, the following are equivalent:

1. $S$48 is u-$S$49-semisimple.
2. Every $S$50-module is u-$S$51-semisimple.
3. Every u-$S$52-short exact sequence is u-$S$53-split.
4. Every short exact sequence is u-$S$54-split.
5. $S$55 is u-$S$56-torsion for all $S$57-modules $S$58.
6. Every $S$59-module is u-$S$60-projective.
7. Every $S$61-module is u-$S$62-injective (Zhang et al., 2021, Zhang et al., 14 Feb 2026).

This theorem is the structural core of the subject. In particular, ring-level u-$S$63-semisimplicity can be viewed equally as a splitting property, an Ext-annihilation property, or universal projectivity/injectivity in the uniform-$S$64 sense. The survey literature also records the equivalent homological-dimension formulation

$S$65

for u-$S$66-semisimple rings (Zhang et al., 14 Feb 2026).

At the module-theoretic level, a later refinement replaces quasi-projective characterizations by pseudo-projective ones. A module $S$67 is u-$S$68-pseudo-projective if, for each submodule $S$69, there exists $S$70 such that any u-$S$71-epimorphism $S$72 lifts after multiplication by $S$73 to an endomorphism of $S$74 (Adarbeh et al., 11 Oct 2025). With this notion, a ring $S$75 is u-$S$76-semisimple if and only if every $S$77-module is u-$S$78-pseudo-projective (Adarbeh et al., 11 Oct 2025). This extends the homological dictionary around u-$S$79-semisimplicity and shows that universal lifting properties persist even after weakening projectivity.

5. Comparison with classical semisimplicity and local criteria

The relation between u-$S$80-semisimplicity and classical semisimplicity depends strongly on the multiplicative subset $S$81.

If $S$82, then u-$S$83-torsion is just zero, u-$S$84-exactness is ordinary exactness, and u-$S$85-splitting is ordinary splitting. In that case, u-$S$86-semisimple modules and rings are exactly classical semisimple modules and rings (Zhang et al., 14 Feb 2026). More generally, if every element of $S$87 is a unit, the uniform-$S$88 theory collapses to the classical one for pseudo-projective and related notions (Adarbeh et al., 11 Oct 2025).

A subtler rigidity theorem holds when $S$89 is regular, meaning that all elements of $S$90 are non-zero-divisors. Then a ring is u-$S$91-semisimple if and only if it is classically semisimple (Zhang et al., 2021, Zhang et al., 14 Feb 2026). Consequently, genuinely new ring-theoretic examples arise only when $S$92 contains zero-divisors or idempotent-like annihilators capable of suppressing non-semisimple components.

There are also local-global criteria. Classical projectivity can be detected by uniform local projectivity: an $S$93-module $S$94 is projective if and only if it is u-$S$95-projective for every prime ideal $S$96, or equivalently for every maximal ideal $S$97 (Zhang et al., 2021). In parallel, a ring $S$98 is classically semisimple if and only if it is u-$S$99-semisimple for every prime ideal $S\subseteq R$00, or equivalently u-$S\subseteq R$01-semisimple for every maximal ideal $S\subseteq R$02 (Zhang et al., 2021). These statements show that the uniform-$S\subseteq R$03 language does not merely relax classical notions; it can also recover them by imposing the uniform condition over all local multiplicative sets.

A related local principle appears for u-$S\subseteq R$04-pseudo-projectivity: if $S\subseteq R$05 is u-$S\subseteq R$06-pseudo-projective for every maximal ideal $S\subseteq R$07, then $S\subseteq R$08 is classically pseudo-projective (Adarbeh et al., 11 Oct 2025). This reinforces the interpretation of uniform-$S\subseteq R$09 notions as local approximants to classical homological properties.

6. Examples, constructions, and structural phenomena

The standard examples clarify both the flexibility and the limits of the theory.

For $S\subseteq R$10 and $S\subseteq R$11, the sequence

$S\subseteq R$12

is u-$S\subseteq R$13-split because the map $S\subseteq R$14 defined by $S\subseteq R$15 satisfies $S\subseteq R$16. Hence $S\subseteq R$17 is u-$S\subseteq R$18-semisimple as a module (Zhang et al., 2021). Nevertheless, $S\subseteq R$19 is not a u-$S\subseteq R$20-semisimple ring, since a u-$S\subseteq R$21-semisimple ring must be uniformly $S\subseteq R$22-von Neumann regular, and $S\subseteq R$23 is not (Zhang et al., 2021).

More generally, if $S\subseteq R$24 is a non-field domain and $S\subseteq R$25, then $S\subseteq R$26 is u-$S\subseteq R$27-semisimple as a module: given a nonzero ideal $S\subseteq R$28, choose $S\subseteq R$29 and define $S\subseteq R$30 by $S\subseteq R$31; then $S\subseteq R$32 for all $S\subseteq R$33, yielding the uniform splitting required by the definition (Zhang et al., 2021). But if $S\subseteq R$34 is regular, such a ring is not u-$S\subseteq R$35-semisimple as a ring unless it is classically semisimple, which a non-field domain is not (Zhang et al., 2021).

The product construction is particularly revealing. If $S\subseteq R$36 and $S\subseteq R$37, then $S\subseteq R$38 is u-$S\subseteq R$39-semisimple if and only if each $S\subseteq R$40 is u-$S\subseteq R$41-semisimple (Zhang et al., 2021, Zhang et al., 14 Feb 2026). Using this, one obtains nonclassical examples: if $S\subseteq R$42 is semisimple, $S\subseteq R$43 is non-semisimple, and

$S\subseteq R$44

then $S\subseteq R$45 is u-$S\subseteq R$46-semisimple although it is not classically semisimple (Zhang et al., 2021, Zhang et al., 14 Feb 2026). The point is that the zero-divisor $S\subseteq R$47 annihilates the obstructing component in the uniform homological conditions.

An example from pseudo-projectivity illustrates the strictness of the hierarchy

$S\subseteq R$48

For $S\subseteq R$49, $S\subseteq R$50, and

$S\subseteq R$51

$S\subseteq R$52 is u-$S\subseteq R$53-pseudo-projective, but not u-$S\subseteq R$54-projective (Adarbeh et al., 11 Oct 2025). This does not directly concern u-$S\subseteq R$55-semisimplicity, but it indicates the breadth of the uniform-$S\subseteq R$56 landscape around the semisimple case.

7. Connections with uniform Artinian theory and later developments

The notion of u-$S\subseteq R$57-semisimplicity is embedded in a larger uniform-$S\subseteq R$58 program that includes Noetherian, Artinian, coherent, regular, flat, injective, and absolutely pure analogues (Zhang et al., 14 Feb 2026). Within this program, u-$S\subseteq R$59-semisimple rings occupy the role of the homological dimension-zero objects, just as semisimple rings do classically.

One major structural interaction is with uniform Artinian theory. A ring $S\subseteq R$60 is u-$S\subseteq R$61-Artinian if there exists $S\subseteq R$62 such that every descending chain of ideals is $S\subseteq R$63-stationary with respect to that fixed $S\subseteq R$64. Any u-$S\subseteq R$65-semisimple ring is u-$S\subseteq R$66-Artinian (Zhang et al., 2022). More generally, $S\subseteq R$67 is u-$S\subseteq R$68-Artinian if and only if $S\subseteq R$69 is u-$S\subseteq R$70-Noetherian, its u-$S\subseteq R$71-Jacobson radical $S\subseteq R$72 is $S\subseteq R$73-nilpotent, and the quotient

$S\subseteq R$74

is a u-$S\subseteq R$75-semisimple ring (Zhang et al., 2022). This is the uniform-$S\subseteq R$76 counterpart of the classical Artinian decomposition via a nilpotent Jacobson radical and semisimple quotient.

The same paper introduces u-$S\subseteq R$77-simple modules, defined so that the module itself is not u-$S\subseteq R$78-torsion but every proper submodule is u-$S\subseteq R$79-torsion with respect to a fixed $S\subseteq R$80. Such modules are u-$S\subseteq R$81-semisimple, and direct sums of arbitrarily many copies of a u-$S\subseteq R$82-simple module remain u-$S\subseteq R$83-semisimple (Zhang et al., 2022). This suggests a decomposition theory partially analogous to classical semisimple decomposition, though the article stops short of a full Wedderburn-type structure theorem for all u-$S\subseteq R$84-semisimple modules.

Recent work further expands the web of equivalent descriptions. Besides the pseudo-projective characterization of u-$S\subseteq R$85-semisimple rings (Adarbeh et al., 11 Oct 2025), the relative projectivity/injectivity framework yields characterizations of u-$S\subseteq R$86-semisimple modules by demanding that all modules be relative u-$S\subseteq R$87-projective or relative u-$S\subseteq R$88-injective against a fixed middle term $S\subseteq R$89 (Adarbeh et al., 1 Sep 2025). The survey literature consolidates these results and places u-$S\subseteq R$90-semisimplicity among the central uniform-$S\subseteq R$91 classes, alongside u-$S\subseteq R$92-von Neumann regular and u-$S\subseteq R$93-Artinian rings (Zhang et al., 14 Feb 2026).

A persistent theme across these developments is that the multiplicative set $S\subseteq R$94 functions as a uniform annihilator of homological defects. When $S\subseteq R$95 is too regular, the classical theory is recovered; when $S\subseteq R$96 contains zero-divisors, one obtains genuinely new semisimple-like phenomena that are invisible in ordinary homological algebra. This suggests that u-$S\subseteq R$97-semisimple modules are best understood not as a mere relaxation of semisimple modules, but as objects in a relative homological theory whose triviality is measured uniformly by $S\subseteq R$98 (Zhang et al., 2021, Zhang et al., 14 Feb 2026).