Properties
One sided primitive rings are both semiprimitive rings and prime rings. Since the ring product of two or more nonzero rings is not prime, it is clear that the product of primitive rings is never primitive.
For a left Artinian ring, it is known that the conditions "left primitive", "right primitive", "prime", and "simple" are all equivalent, and in this case it is a semisimple ring isomorphic to a square matrix ring over a division ring. More generally, in any ring with a minimal one sided ideal, "left primitive"="right primitive"="prime".
A commutative ring is left primitive if and only if it is a field.
Being left primitive is a Morita invariant property.
Read more about this topic: Primitive Ring
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)