On flat factor rings and fully right idempotent rings

Baccella, Giuseppe

doi:10.1007/bf02825175

Cited by 10 publications

(1 citation statement)

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“…A ring R is called fully right idempotent if I = I 2 , for every right ideal I. We recall that a right V-ring is fully right idempotent (see [19,Corollary 2.2]) and a prime fully right idempotent ring is right non-singular (see [2,Lemma 4.3]). So a prime right V-ring is right non-singular.…”

Section: Definitions and Notationmentioning

confidence: 99%

Indecomposable decomposition and couniserial dimension

2014

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Dimensions like Gelfand, Krull, Goldie have an intrinsic role in the study of theory of rings and modules. They provide useful technical tools for studying their structure. We define and study couniserial dimension for modules. Couniserial dimension is a measure of how far a module deviates from being uniform. Despite their different objectives, it turns out that there are certain common properties between the couniserial dimension and Krull dimension. Among others, each module having such a dimension contains a uniform submodule and has finite uniform dimension. Like all dimensions, this is an ordinal valued invariant. Every module of finite length has couniserial dimension and its value lies between the uniform dimension and the length of the module. Modules with countable couniserial dimension are shown to possess indecomposable decomposition. In particular, a von Neumann regular ring with countable couniserial dimension is semisimple artinian. If the maximal right quotient ring of a semiprime right non-singular ring R has a couniserial dimension as an R-module, then R is a semiprime right Goldie ring. As one of the applications, it follows that all right R-modules have couniserial dimension if and only if R is a semisimple artinian ring.

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Section: Definitions and Notationmentioning

confidence: 99%