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Essentially compressible modules and rings. (English) Zbl 1114.16007

J. Algebra 304, No. 2, 812-831 (2006); Corrigendum 310, No. 1, 458 (2007).
Let \(R\) be a ring with identity element. A unital right \(R\)-module \(M\) is called ‘essentially compressible’ if \(M\) embeds in every essential submodule of \(M\). The ring \(R\) is said to be ‘right essentially compressible’ if the right module \(R_R\) is essentially compressible.
The aim of this paper is a study of essentially compressible modules and rings. Thus, the authors prove that every non-singular essentially compressible \(R\)-module is isomorphic to a submodule of a free \(R\)-module, and the converse holds if \(R\) is semiprime right Goldie. In case \(R\) is a right FBN ring, it is shown that a right \(R\)-module \(M\) is essentially compressible if and only if \(M\) embeds in a direct sum of critical compressible right \(R\)-modules. Then, the authors characterize the right essentially compressible rings, and as a consequence, it follows that \(R\) is semiprime right Goldie if and only if \(R\) is a right essentially compressible ring having at least one uniform right ideal.

MSC:

16D80 Other classes of modules and ideals in associative algebras
16P60 Chain conditions on annihilators and summands: Goldie-type conditions
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