Primitive ring
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In abstract algebra, a left primitive ring R is a ring with a faithful simple left module R-module. A right primitive ring is defined similarly.
Primitive rings generalize simple rings.
[edit] Properties
- Every simple ring is primitive. This may not hold in the case that we do not assume rings to have multiplicative identities.
- An Artinian ring is primitive if and only if it is simple.
- A commutative ring is primitive if and only if it is a field.
As a consequence of the Jacobson Density Theorem, every primitive ring is a dense subring of the ring of linear transformations of a vector space over a division ring. For left primitive rings, these linear transformations act on the left, and for right primitive rings, they act on the right.
Conversely, it is easy to see that every dense subring of the ring of linear transformations of a vector space over a division ring is primitive. Thus, the theorem completely characterizes primitive rings.
A ring theoretic formulation of a left primitive ring is as follows : there is a maximal left ideal whose right core is zero. The dual definition is valid for right primitive rings.
[edit] Nice facts
- There are primitive rings which are not simple. In particular, the ring of all linear transformations of an infinite dimensional vector space over a division ring is primitive, but is not simple as the set of finite rank linear transformations is a two sided ideal.
- There are left primitive rings which are not right primitive.
