Answer to Question #16361 in Abstract Algebra for Melvin Henriksen
Does every simple ring have IBN?
The answer is "no." Take any nonzero ring R which does not have IBN. Let I be a maximal ideal of R. Then S = R/ I is a simple ring, and we have a surjection R->S . Since R does not have IBN, it follows that S also does not have IBN.