Cotorsion Modules and Pure-Injective Envelopes

Definitions and mathematical statement

Let $R$ be an associative unital ring, not assumed commutative, and work with unital left modules. A left module $C$ is cotorsion if \[ \operatorname{Ext}_R^1(F,C)=0\qquad\text{for every flat left }R\text{-module }F. \] An exact sequence of left modules is pure if it remains exact after tensoring with every right module. A left module $E$ is pure-injective if maps into $E$ extend across pure monomorphisms. The first source question asks for a ring $R$ that is noncoherent but satisfies \[ \forall C\in R\text{-}\operatorname{Mod},\qquad C\text{ cotorsion}\ \Longrightarrow\ C\text{ pure-injective}. \] The quantifier is over all left modules, not only finitely generated ones. For a noncommutative ring, right coherence means every finitely generated right ideal is finitely presented, and left coherence is the analogous left-sided condition. The source does not specify which coherence fails; a side cannot be selected silently from the left-module wording.

For a left module $M$, let $i_M:M\hookrightarrow\operatorname{PE}(M)$ be its pure-injective envelope, with its usual pure embedding and minimal-envelope property, and put \[ C_M=\operatorname{coker}i_M=\operatorname{PE}(M)/i_M(M). \] The second question asks for ring-theoretic properties equivalent to \[ \forall M\in R\text{-}\operatorname{Mod},\qquad C_M\text{ is pseudoflat}. \] The source supplies no definition of pseudoflat and no proposed list of equivalent properties. In particular it is not replaced by flatness, a guessed Tor-vanishing condition, or a finitely-presented approximation property. The relevant side convention for pseudoflatness must also be fixed. This named-property gap prevents a self-contained characterization theorem for the second question.

The two requests are retained separately; the record does not assert that the cotorsion condition and the envelope-cokernel condition are themselves equivalent. The supplied literature triage checked 2026-08-21 reports no authoritative complete resolution of the extracted pair, without a specific example, characterization, or pseudoflatness definition.

Short English statement

Seek a noncoherent ring whose cotorsion left modules are all pure-injective, and characterize rings whose every pure-injective-envelope cokernel is pseudoflat. The coherence side and the definition of pseudoflat are essential missing data.

Sources

• Local UnsolvedMath registry, permanent ID 30000678, catalogue entry, Cotorsion Modules and Pure-Injective Envelopes.
• Model Theory and Groups (2007), Oberwolfach Reports, pp. 83--138, DOI https://doi.org/10.4171/owr/2007/02.
• Supplied source-verification claim and literature triage checked 2026-08-21 cite only the report; no separate ring characterization, example, or pseudoflatness definition is supplied.