semiprimitive ring SemiprimitiveRing 2002-04-20 02:57:53 2006-09-27 13:48:12 Definition semiprimitivity semiprimitive semisimple Jacobson semisimple J-semisimple semi-primitivity semi-primitive semi-simple Jacobson semi-simple J-semi-simple \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \PMlinkescapeword{states} A ring is said to be \emph{semiprimitive} if its Jacobson radical is the zero ideal. Any simple ring is automatically semiprimitive. A finite direct product of matrix rings over division rings can be shown to be semiprimitive and both left and right Artinian. The \PMlinkname{Artin-Wedderburn Theorem}{WedderburnArtinTheorem} states that any semiprimitive ring which is left or right Artinian is isomorphic to a finite direct product of matrix rings over division rings. {\bf Note}: The semiprimitive condition is sometimes also referred to as a \emph{semisimple}, \emph{Jacobson semisimple}, or \emph{J-semisimple}. Furthermore, when either of the last two names are used, the adjective 'semisimple' is frequently intended to refer to a ring that is semiprimitive and Artinian (see the entry on \PMlinkname{semisimple rings}{SemisimpleRing2}).