Simple Ring

In abstract algebra, a simple ring is a non-zero ring that has no ideal besides the zero ideal and itself. A simple ring can always be considered as a simple algebra. According to the Artin-Wedderburn theorem, every simple ring that is left or right Artinian is a matrix ring over a division ring. In particular, the only simple rings that are a finite-dimensional vector space over the real numbers are rings of matrices over either the real numbers, the complex numbers, or the quaternions. An example of a simple ring that is not a matrix ring over a division ring is the Weyl algebra. Any quotient of a ring by a maximal ideal is a simple ring.

 

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