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Encyclopedia > 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. Results from FactBites:

 Simple aromatic ring - Wikipedia, the free encyclopedia (379 words) Simple aromatic rings are aromatic organic compounds (also known as arenes or aromatics) that consist only of conjugated planar ring systems with delocalized pi electron clouds instead of discrete alternating single and double bonds. Typical simple aromatic compounds are benzene and indole. In the oxygen (O) and sulfur (S) containing aromatic rings one of the electron pairs of the heteroatoms contributes to the aromatic system (similar to the non-basic nitrogen containing rings), while the second lone pair extends in the plane of the ring (similar to the basic nitrogen containing rings).
 Simple ring - Wikipedia, the free encyclopedia (161 words) In abstract algebra, a simple ring is a non-zero ring that has no ideal besides the zero ideal and itself. 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.
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