Sentence examples for modules have couniserial from inspiring English sources

If all finitely generated right modules have couniserial dimension, then every right module contains a noetherian uniform module.

(3) All left (R -modules have couniseR -modulession.

(2) All right (R -modules have couniseR -modulession.

As one of the applications, it follows that all right (R -modules have couniseR -modulession if and only if (R) is a semisimple artinian ring.

As another application we show that all right (left) (R -module have couniseR -modulension if and only if (R) is semisimple artinian (see Thaveem 5.8).

Every artinian module has couniserial dimension.

If (M) as (R -module has couniseR -modulension, thas (M) as (R/I)-module has couniserial dimension.

If (S oplus D) as (D -module has couniserial D -module, thas (D) is a princouniserial idimensionin.

If (Q) as an (R -module has couniseR -modulension, thas (R) is a semiprime right Goldie ring whicouniserialite prodimensionrime Goldie rings, each of which is a piecewise domain.

Indeed it follows from Theorem 4.10, proved later that, if the maximal right quotient ring (Q) of a domain (D) has couniserial dimension as (D -module, then (Q_D -modulencellathen property and so if (Q_Dplus D) as (D)-module has cancellationdimension, (D) must be right propertyl ideanddomain.

The next proposition provides a working criterion for a module to have couniserial dimension.