Jacobsons Lemma fails for nil-clean 2 × 2 integral matrices

Abstract

We show that for two 2 × 2 integral matrices A, B, if the product AB is nil-clean then BA may not be nil-clean. Despite the fact that for many special cases, BA is also nil-clean, we finally found three counterexamples. All the way, the computer aid was decisive.

Keywords

• nil-clean
• clean
• 2 × 2 matrix
• principal ideal domain
• primary 16u10, 16u60
• secondary 15-04, 16-04, 15b36