In abstract algebra, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. A simple ring can always be considered as a simple algebra. Rings which are simple as rings but not as modules do exist: the full matrix ring over a field does not have any nontrivial ideals (since any ideal of M(n,R) is of the form M(n,I) with I an ideal of R), but has nontrivial left ideals (namely, the sets of matrices which have some fixed zero columns).
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.
Any quotient of a ring by a maximal ideal is a simple ring. In particular, a field is a simple ring. A ring R is simple if and only its opposite ring Ro is simple.
An example of a simple ring that is not a matrix ring over a division ring is the Weyl algebra.
Read more about Simple Ring: Wedderburn's Theorem
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