<i> δ <sub>ss</sub> </i> -supplemented modules and rings
Burcu Nişancı Türkmen, Ergül Türkmen
Abstract
Abstract In this paper, we introduce the concept of δ ss -supplemented modules and provide the various properties of these modules. In particular, we prove that a ring R is δ ss -supplemented as a left module if and only if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mfrac> <m:mi>R</m:mi> <m:mrow> <m:mi>S</m:mi> <m:mi>o</m:mi> <m:mi>c</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow/> <m:mi>R</m:mi> </m:msub> <m:mi>R</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:mfrac> </m:mrow> </m:math> {R \over {Soc\left( {_RR} \right)}} is semisimple and idempotents lift to Soc ( R R ) if and only if every left R -module is δ ss -supplemented. We define projective δ ss -covers and prove the rings with the property that every (simple) module has a projective δ ss -cover are δ ss -supplemented. We also study on δ ss -supplement submodules.