Ring theory

Simple ring

In abstract algebra, a branch of mathematics, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. In particular, a commutative ring is a simple ring if and only if it is a field. The center of a simple ring is necessarily a field. It follows that a simple ring is an associative algebra over this field. So, simple algebra and simple ring are synonyms. Several references (e.g., Lang (2002) or Bourbaki (2012)) require in addition that a simple ring be left or right Artinian (or equivalently semi-simple). Under such terminology a non-zero ring with no non-trivial two-sided ideals is called quasi-simple. Rings which are simple as rings but are not a simple module over themselves do exist: a full matrix ring over a field does not have any nontrivial ideals (since any ideal of is of the form with an ideal of ), but has nontrivial left ideals (for example, 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. An example of a simple ring that is not a matrix ring over a division ring is the Weyl algebra. (Wikipedia).

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Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Definition of a Ring and Examples of Rings - Definition of a Ring. - Definition of a commutative ring and a ring with identity. - Examples of Rings include: Z, Q, R, C under regular addition and multiplication The Ring of all n x

From playlist Abstract Algebra

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From playlist Abstract Algebra

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From playlist Abstract Algebra

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From playlist Abstract Algebra

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From playlist Visual Group Theory

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From playlist Rings and modules

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From playlist Rings and modules

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From playlist Abstract Algebra

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From playlist Abstract Algebra

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From playlist Commutative algebra

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From playlist Chemistry 202. Organic Reaction Mechanisms II

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From playlist Representation theory

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From playlist Global Noncommutative Geometry Seminar (Europe)

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From playlist 2022 Summer School on the Langlands program

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From playlist Visual Group Theory

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From playlist Abstract Algebra

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Related pages

Semisimple algebra | Finite field | Division ring | Associative algebra | Center (ring theory) | Ideal (ring theory) | Krull dimension | Zero ring | Isomorphism | Quaternion | Artinian ring | Frobenius theorem (real division algebras) | Module homomorphism | Matrix ring | Idempotent matrix | Mathematics | Field (mathematics) | Simple module | Real number | Noetherian ring | Reduced ring | Ring (mathematics) | Weyl algebra | Abstract algebra | Complex number | Division algebra | Central simple algebra | Commutative ring