A ring is simple if it is it is a simple object in the category of --bimodules.
This can be stated in more elementary terms in any of the following equivalent ways:
is nontrivial and has no nontrivial two-sided ideals.
has exactly two two-sided ideals (which must be itself and its zero ideal).
In constructive algebra, this is too strong; we must say:
A field, or a division ring, is simple.
If is a division ring, then the ring of matrices with entries in is a simple ring.
The Weyl algebra over a field is a simple ring. (In different language: this is the ring of differential operators with polynomial coefficients in one variable , obtained as the image of the ring homomorphism from the noncommutative polynomial ring to the ring of -linear endomorphisms that sends to the derivative operator and to the multiplication operator .) An explanation of why this is simple may be found here at Qiaochu Yuan‘s blog.
Last revised on September 2, 2024 at 14:32:21. See the history of this page for a list of all contributions to it.