If the left annihilator of any principal left ideal of End_R ( M ) is a direct summand of M then M is called left principally quasi-Baer .

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( I ) Every maximal direct summand of M has unique complement .

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Is generalized essential finitely generated - module if and only if for any generalized essential finitely generated sub-module there exists a type direct summand of such that N u2264 geK .

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The Discussion of the Indecomposable Direct Summand of the Induced Module From u03b8 - Module

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In this paper we make some discussions in the central problem of the cancellation property of quasi discrete modules and obtain the following main results : ( 1 ) M has the lifting property if and only if any co closed submodule is a summand of M ;

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A ring R is called left ( right ) . G-regular if each finitely presented submodule of a free left ( right ) R - module is a summand .

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It proves that the direct summand of a weak CS-module is not a weak CS-module by using a counter example and that the weak CS-module satisfying the C_3-condition is of the hereditary property for its direct summand .

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Adjoint clifford rings and the rings with small circle semigroups are characterized in terms of circle multiplication and a ( commutative ) adjoint Clifford ring is constructed in which Jacobson radical is not a summand so as to confirm a conjecture of ( Heatherly ) and Tucci .

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