Artinian ring
A simple Artinian ring "A" is a matrix ring over a division ring. Indeed, let "I" be a minimal (nonzero) right ideal of "A". Then, since formula_11 is a two-sided ideal, formula_12 since "A" is simple. Thus, we can choose formula_13 so that formula_14. Assume "k" is minimal with respect that property. Consider the map of right "A"-modules: