Examples
Every simple ring R with unity is both left and right primitive. (However, a simple non-unital ring may not be primitive.) This follows from the fact that R has a maximal left ideal M, and the fact that the quotient module R/M is a simple left R-module, and that its annihilator is a proper two-sided ideal in R. Since R is a simple ring, this annihilator is {0} and therefore R/M is a faithful left R-module.
Weyl algebras over fields with characteristic zero are primitive, and since they are domains, they are examples without minimal one-sided ideals.
Read more about this topic: Primitive Ring
Famous quotes containing the word examples:
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)
“In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
—Michel de Montaigne (1533–1592)