@article{article_1058427, title={On I-finite left quasi-duo rings}, journal={International Electronic Journal of Algebra}, volume={31}, pages={161–202}, year={2022}, DOI={10.24330/ieja.1058427}, author={Horoub, Ayman M. A. and Nıcholson, W. K.}, keywords={Left quasi-duo ring, left soclin ring, generalized triangular matrix ring, left-max ideal, ideal-simple module, left socle, very semisimple module}, abstract={<p>A ring is called left quasi-duo (left QD) if every maximal left ideal is a
right ideal, and it is called I-finite if it contains no infinite orthogonal
set of idempotents. It is shown that a ring is I-finite and left QD if and
only if it is a generalized upper-triangular matrix ring with all diagonal
rings being division rings except the lower one, which is either a division
ring or it is I-finite, left QD and left `soclin’ (left QDS). Here a ring is
called left soclin if each simple left ideal is nilpotent. The left QDS
rings are shown to be finite direct products of indecomposable left QDS
rings, in each of which <span class="MathJax_Preview" style="color:inherit;"> </span> <span class="mjx-chtml MathJax_CHTML" style="font-size:123%;"> <span class="mjx-math"> <span class="mjx-mrow"> <span class="mjx-mn"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.378em;padding-bottom:.316em;">1 </span> </span> <span class="mjx-mo MJXc-space3"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.066em;padding-bottom:.316em;">= </span> </span> <span class="mjx-msubsup MJXc-space3"> <span class="mjx-base" style="margin-right:-.06em;"> <span class="mjx-mi"> <span class="mjx-char MJXc-TeX-math-I" style="padding-top:.503em;padding-bottom:.503em;padding-right:.06em;">f </span> </span> </span> <span class="mjx-sub" style="font-size:70.7%;vertical-align:-.212em;padding-right:.071em;"> <span class="mjx-texatom"> <span class="mjx-mrow"> <span class="mjx-mn"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.378em;padding-bottom:.316em;">1 </span> </span> </span> </span> </span> </span> <span class="mjx-mo MJXc-space2"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.316em;padding-bottom:.441em;">+ </span> </span> <span class="mjx-mo MJXc-space2"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.003em;padding-bottom:.316em;">⋯ </span> </span> <span class="mjx-mo MJXc-space2"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.316em;padding-bottom:.441em;">+ </span> </span> <span class="mjx-msubsup MJXc-space2"> <span class="mjx-base" style="margin-right:-.06em;"> <span class="mjx-mi"> <span class="mjx-char MJXc-TeX-math-I" style="padding-top:.503em;padding-bottom:.503em;padding-right:.06em;">f </span> </span> </span> <span class="mjx-sub" style="font-size:70.7%;vertical-align:-.212em;padding-right:.071em;"> <span class="mjx-texatom"> <span class="mjx-mrow"> <span class="mjx-mi"> <span class="mjx-char MJXc-TeX-math-I" style="padding-top:.191em;padding-bottom:.316em;">m </span> </span> </span> </span> </span> </span> </span> </span> <span class="MJX_Assistive_MathML">1=f1+⋯+fm </span> </span> where the <span class="MathJax_Preview" style="color:inherit;"> </span> <span class="mjx-chtml MathJax_CHTML" style="font-size:123%;"> <span class="mjx-math"> <span class="mjx-mrow"> <span class="mjx-msubsup"> <span class="mjx-base" style="margin-right:-.06em;"> <span class="mjx-mi"> <span class="mjx-char MJXc-TeX-math-I" style="padding-top:.503em;padding-bottom:.503em;padding-right:.06em;">f </span> </span> </span> <span class="mjx-sub" style="font-size:70.7%;vertical-align:-.212em;padding-right:.071em;"> <span class="mjx-texatom"> <span class="mjx-mrow"> <span class="mjx-mi"> <span class="mjx-char MJXc-TeX-math-I" style="padding-top:.441em;padding-bottom:.316em;">i </span> </span> </span> </span> </span> </span> </span> </span> <span class="MJX_Assistive_MathML">fi </span> </span> are
orthogonal primitive idempotents, with <span class="MathJax_Preview" style="color:inherit;"> </span> <span class="mjx-chtml MathJax_CHTML" style="font-size:123%;"> <span class="mjx-math"> <span class="mjx-mrow"> <span class="mjx-msubsup"> <span class="mjx-base" style="margin-right:-.06em;"> <span class="mjx-mi"> <span class="mjx-char MJXc-TeX-math-I" style="padding-top:.503em;padding-bottom:.503em;padding-right:.06em;">f </span> </span> </span> <span class="mjx-sub" style="font-size:70.7%;vertical-align:-.219em;padding-right:.071em;"> <span class="mjx-texatom"> <span class="mjx-mrow"> <span class="mjx-mi"> <span class="mjx-char MJXc-TeX-math-I" style="padding-top:.441em;padding-bottom:.316em;">k </span> </span> </span> </span> </span> </span> <span class="mjx-mo MJXc-space3"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.191em;padding-bottom:.316em;">≈ </span> </span> <span class="mjx-msubsup MJXc-space3"> <span class="mjx-base" style="margin-right:-.06em;"> <span class="mjx-mi"> <span class="mjx-char MJXc-TeX-math-I" style="padding-top:.503em;padding-bottom:.503em;padding-right:.06em;">f </span> </span> </span> <span class="mjx-sub" style="font-size:70.7%;vertical-align:-.219em;padding-right:.071em;"> <span class="mjx-texatom"> <span class="mjx-mrow"> <span class="mjx-mi"> <span class="mjx-char MJXc-TeX-math-I" style="padding-top:.441em;padding-bottom:.316em;">l </span> </span> </span> </span> </span> </span> </span> </span> <span class="MJX_Assistive_MathML">fk≈fl </span> </span> for all <span class="MathJax_Preview" style="color:inherit;"> </span> <span class="mjx-chtml MathJax_CHTML" style="font-size:123%;"> <span clas}, number={31}, publisher={Abdullah HARMANCI}