semiprimitive ring
A ring is said to be 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 Artin-Wedderburn Theorem (http://planetmath.org/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.
Note:
The semiprimitive condition is sometimes also referred to as a semisimple, Jacobson semisimple, or 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 semisimple rings
(http://planetmath.org/SemisimpleRing2)).
| Title | semiprimitive ring |
| Canonical name | SemiprimitiveRing |
| Date of creation | 2013-03-22 12:36:14 |
| Last modified on | 2013-03-22 12:36:14 |
| Owner | yark (2760) |
| Last modified by | yark (2760) |
| Numerical id | 20 |
| Author | yark (2760) |
| Entry type | Definition |
| Classification | msc 16N20 |
| Synonym | semisimple ring |
| Synonym | Jacobson semisimple ring |
| Synonym | J-semisimple ring |
| Synonym | semi-primitive ring |
| Synonym | semi-simple ring |
| Synonym | Jacobson semi-simple ring |
| Synonym | J-semi-simple ring |
| Related topic | SemisimpleRing2 |
| Related topic | WedderburnArtinTheorem |
| Defines | semiprimitivity |
| Defines | semiprimitive |
| Defines | semisimple |
| Defines | Jacobson semisimple |
| Defines | J-semisimple |
| Defines | semi-primitivity |
| Defines | semi-primitive |
| Defines | semi-simple |
| Defines | Jacobson semi-simple |
| Defines | J-semi-simple |