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Let $R$ be a ring with identity and let $M$ be a unitary right
$R$-module. Then, $M$ is essentially compressible provided $M$
embeds in every essential submodule of $M$. It is proved that
every nonsingular essentially compressible module $M$ is
isomorphic to a submodule of a free module, and the converse holds
in case $R$ is semiprime right Goldie. In case $R$ is a right FBN
ring, $M$ is essentially compressible if and only if $M$ is
subisomorphic to a direct sum of critical compressible modules.
The ring $R$ is right essentially compressible if and only if
there exist a positive integer $n$ and prime ideals $P_{i}(1\leq i
\leq n)$ such that $P_{1}\cap...\cap P_{n}=0$ and the prime ring
$R/P_{i}$ is right essentially compressible for each $1\leq i \leq
n$. It follows that a ring $R$ is semiprime right Goldie if and
only if $R$ is a right essentially compressible ring with at least
one uniform right ideal.
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