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A ring $R$ is said to be {\em semiprimitive} if its Jacobson radical is the zero ideal.\\ |
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A ring $R$ is said to be {\em semiprimitive} if its Jacobson radical is the zero ideal.
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Any simple ring is automatically semiprimitive.\\
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| 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.\\ |
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| A finite direct product of matrix rings over division rings can be shown to be semiprimitive and both left and right Artinian. |
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| The Artin-Wedderburn Theorem states that any semiprimitive ring which is left or right Artinian is isomorphic to a finite direct product of matrix rings over division rings. |
The Artin-Wedderburn Theorem states that any semiprimitive ring which is left or right Artinian is isomorphic to a finite direct product of matrix rings over division rings. |
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| {\em Note:} |
{\em Note:} |
| The 'semiprimitive' condition is sometimes also referred to as a {\em semisimple}, {\em Jacobson semisimple}, or {\em J-semisimple}. Furthermore, when either of the latter two names are used, the adjective 'semisimple' is frequently intended to refer to a ring that is semiprimitive and Artinian. |
The 'semiprimitive' condition is sometimes also referred to as a {\em semisimple}, {\em Jacobson semisimple}, or {\em J-semisimple}. Furthermore, when either of the latter two names are used, the adjective 'semisimple' is frequently intended to refer to a ring that is semiprimitive and Artinian. |
