TY - JOUR AU - Barros, Constantino M. de PY - 1968/01/01 Y2 - 2024/03/29 TI - Une caracterisation des anneaux fortemen réguliers JF - Revista Colombiana de Matemáticas JA - rev.colomb.mat VL - 2 IS - 1 SE - Artículos DO - UR - https://revistas.unal.edu.co/index.php/recolma/article/view/31450 SP - 1-5 AB - <p style="margin-bottom: 0.0001pt;"><span style="font-size: 10pt; font-family: ";Arial";,";sans-serif";;">On montre que la classe des anneaux fortement réguliers introduits et étudies par Arens-Kaplansky [1] coincide avec celle des anneaux dont le demi-groupe multiplicatif est inverse, donc coincide avec celle des anneaux réguliers dont l'ensemble de leurs idempotents est commutatif.</span></p> <p style="margin-bottom: 0.0001pt;"><span style="font-size: 10pt; font-family: ";Arial";,";sans-serif";;">1. Soit A un anneau. Si A possede un unique élément unité à droite e, alors e est aussi une unite à gaucne. En effet, soit U</span><sub><span style="font-size: 8pt; font-family: ";Arial";,";sans-serif";;">d</span></sub><span style="font-size: 10pt; font-family: ";Arial";,";sans-serif";;">(A) l'ensemble des elements unites à droite de A. Pour chaque e (pertenece)</span><span style="font-family: Symbol;"></span> <span style="font-size: 10pt; font-family: ";Arial";,";sans-serif";;">U</span><sub><span style="font-size: 8pt; font-family: ";Arial";,";sans-serif";;">d</span></sub><sub><span style="font-size: 10pt; font-family: ";Arial";,";sans-serif";;"><span>  </span></span></sub><span style="font-size: 10pt; font-family: ";Arial";,";sans-serif";;">(A), soit P</span><sub><span style="font-size: 8pt; font-family: ";Arial";,";sans-serif";;">e </span></sub><sub><span style="font-size: 10pt; font-family: ";Arial";,";sans-serif";;"><span> </span></span></sub><span style="font-size: 10pt; font-family: ";Arial";,";sans-serif";;">l'applica tion de <span> </span>A <span> </span>dans <span> </span>A <span> </span>telle que </span></p> <p style="margin-bottom: 0.0001pt;"><span style="font-size: 10pt; font-family: ";Arial";,";sans-serif";;">P</span><sub><span style="font-size: 8pt; font-family: ";Arial";,";sans-serif";;">e </span></sub><span style="font-size: 10pt; font-family: ";Arial";,";sans-serif";;">(x) = ex - x + e. On a<span>   </span><span> </span></span></p> <p style="margin-bottom: 0.0001pt;"><span style="font-size: 10pt; font-family: ";Arial";,";sans-serif";;">(a) P</span><sub><span style="font-size: 8pt; font-family: ";Arial";,";sans-serif";;">e</span></sub><span style="font-size: 8pt; font-family: ";Arial";,";sans-serif";;"> </span><span style="font-size: 10pt; font-family: ";Arial";,";sans-serif";;">(A) (inclusión) U</span><sub><span style="font-size: 8pt; font-family: ";Arial";,";sans-serif";;">d</span></sub><span style="font-size: 10pt; font-family: ";Arial";,";sans-serif";;"> (A). En effet, pour tout y  <span>(pertenece)</span></span><span style="font-family: Symbol;"></span><span style="font-size: 10pt; font-family: ";Arial";,";sans-serif";;"> A on a <span>  </span>y </span><sup><span style="font-size: 11pt; font-family: ";Arial";,";sans-serif";;">P</span></sup><span style="font-size: 8pt; font-family: ";Arial";,";sans-serif";;">e </span><span style="font-size: 10pt; font-family: ";Arial";,";sans-serif";;">(x) = y,<span>  </span></span></p><p style="margin-bottom: 0.0001pt;"><span style="font-size: 10pt; font-family: ";Arial";,";sans-serif";;"><span> (b) </span>la restriction de P</span><sub><span style="font-size: 8pt; font-family: ";Arial";,";sans-serif";;">e</span></sub><span style="font-size: 10pt; font-family: ";Arial";,";sans-serif";;"> a U</span><sub><span style="font-size: 8pt; font-family: ";Arial";,";sans-serif";;">d</span></sub><sub><span style="font-size: 10pt; font-family: ";Arial";,";sans-serif";;"> </span></sub><span style="font-size: 10pt; font-family: ";Arial";,";sans-serif";;">(A) <span> </span>est injestive. </span></p> <p style="margin-bottom: 0.0001pt;"><span style="font-size: 10pt; font-family: ";Arial";,";sans-serif";;">En effet, soient <span> </span>e´, e" (pertenece) </span><span style="font-family: Symbol;"></span><span style="font-size: 10pt; font-family: ";Arial";,";sans-serif";;"> U</span><sub><span style="font-size: 8pt; font-family: ";Arial";,";sans-serif";;">d</span></sub><span style="font-size: 10pt; font-family: ";Arial";,";sans-serif";;"> (A), alors P</span><sub><span style="font-size: 8pt; font-family: ";Arial";,";sans-serif";;">e</span></sub><sub><span style="font-size: 14pt; font-family: ";Arial";,";sans-serif";;"> </span></sub><span style="font-size: 10pt; font-family: ";Arial";,";sans-serif";;">(e') = P</span><sub><span style="font-size: 8pt; font-family: ";Arial";,";sans-serif";;">e</span></sub><sub><span style="font-size: 11pt; font-family: ";Arial";,";sans-serif";;"> </span></sub><span style="font-size: 10pt; font-family: ";Arial";,";sans-serif";;">(ee") entraíne </span></p><p style="margin-bottom: 0.0001pt;"><span style="font-size: 10pt; font-family: ";Arial";,";sans-serif";;">ee´ - e´+ e = ee" – ee" + e, </span></p><p style="margin-bottom: 0.0001pt;"><span style="font-size: 10pt; font-family: ";Arial";,";sans-serif";;">mais ee´ e = ee" donc <span> </span>e´ = ee". <span> </span></span></p><p style="margin-bottom: 0.0001pt;"><span style="font-size: 10pt; font-family: ";Arial";,";sans-serif";;"></span><span style="font-size: 10pt; font-family: ";Arial";,";sans-serif";;">Suppasons U</span><sub><span style="font-size: 8pt; font-family: ";Arial";,";sans-serif";;">d</span></sub><sub><span style="font-size: 14pt; font-family: ";Arial";,";sans-serif";;"> </span></sub><span style="font-size: 10pt; font-family: ";Arial";,";sans-serif";;">(A) = e. D'aprés <span> </span>(a) <span> </span>pour tout <span> </span></span></p><p style="margin-bottom: 0.0001pt;"><span style="font-size: 10pt; font-family: ";Arial";,";sans-serif";;">x (pertenece)</span><span style="font-family: Symbol;"></span><span style="font-size: 10pt; font-family: ";Arial";,";sans-serif";;"> <span>A on a P</span></span><sub><span style="font-size: 8pt; font-family: ";Arial";,";sans-serif";;">e</span></sub><sub><span style="font-size: 14pt; font-family: ";Arial";,";sans-serif";;"> </span></sub><span style="font-size: 10pt; font-family: ";Arial";,";sans-serif";;">(x) = e,  c´ est-à.-dire <span> </span>ex = x. <span> </span>Par conséquent e est un élément unité de A.</span></p> ER -