Strongly J-Clean Matrices Over Local Rings

An element of a ring is called strongly J-clean provided that it can be written as the sum of an idempotent and an element in its Jacobson radical that commute. We investigate, in this article, a single strongly J-clean 2 × 2 matrix over a noncommutative local ring. The criteria on strong J-cleanness of 2 × 2 matrices in terms of a quadratic equation are given. These extend the corresponding results in 8 Li , B. ( 2009 ). Strongly clean matrix rings over noncommutative local rings . Bull. Korean Math. Soc. 46 : 71 – 78 .Crossref], Web of Science ®] , Google Scholar], Theorems 2.7 and 3.2], 9 Li , Y. ( 2007 ). Strongly clean matrix rings over local rings . J. Algebra 312 : 397 – 404 .Crossref], Web of Science ®] , Google Scholar], Theorem 2.6], and 11 Yang , X. , Zhou , Y. ( 2008 ). Strongly cleanness of the 2 × 2 matrix ring over a general local ring . J. Algebra 320 : 2280 – 2290 .Crossref], Web of Science ®] , Google Scholar], Theorem 7].