nLab
simple ring

A ring RR is simple if it is it is a simple object in the category of RR-RR-bimodules.

This can be stated in more elementary terms in any of the following equivalent ways:

  • RR is nontrivial and has no nontrivial two-sided ideals.
  • RR has exactly two two-sided ideals (which must be RR itself and its zero ideal).

In constructive algebra, this is too strong; we must say:

  • For each two-sided ideal II, II is the zero ideal if and only if II is proper (not equal to RR).

Examples

  • A field, or a division ring, is simple.

  • If DD is a division ring, then the ring M n(D)M_n(D) of n×nn \times n matrices with entries in DD is a simple ring.

  • The Weyl algebra kx,y/(xyyx1)k\langle x, y\rangle/(x y - y x - 1) over a field kk is a simple ring. (In different language: this is the ring of differential operators with polynomial coefficients in one variable tt, obtained as the image of the ring homomorphism from the noncommutative polynomial ring kx,yk \langle x, y \rangle to the ring of kk-linear endomorphisms Vect(k[t],k[t])Vect(k[t], k[t]) that sends xx to the derivative operator ddt\frac{d}{d t} and yy to the multiplication operator tt \cdot -.) An explanation of why this is simple may be found here at Qiaochu Yuan’s blog.

Revised on February 5, 2016 20:38:42 by Todd Trimble (67.81.95.215)