This can be stated in more elementary terms in any of the following equivalent ways:
In constructive algebra, this is too strong; we must say:
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.