Simple ring

Symbol-free definition
A ring is termed simple if it satisfies the following equivalent conditions:


 * It has no proper nontrivial two-sided ideal
 * Any homomorphism from it is either trivial or injective

Stronger properties

 * Field
 * Division ring
 * Skew field

Weaker properties

 * Semisimple ring
 * Left primitive ring
 * Right primitive ring

Left-right symmetry
The property of being a simple ring is left-right symmetric. That is, a ring is simple if and only if its opposite ring is simple.