Strongly Clean Matrices over Commutative Rings

On strongly clean matrices over commutative clean rings

“…Proof. Theorem 2.4 of [8] gives the equivalence of the three statements for projective free rings and the proof of Theorem 7 shows how this lifts to the ring R.…”

“…The key ingredients of [2, Proposition 44], which is about local rings, are the fact that a local ring is projective free and that an SP-factorization (which is clearly an SRfactorization) is, in fact a SRC-factorization. However, the quoted lemma from [8] makes it clear that if R is a commutative projective free ring and h = h 0 p 0 is an SP-factorization of a monic h ∈ R[t] then (h 0 , p 0 ) = R[t]. The result yields a more general form of Proposition 14.…”

“…In [8] the authors generalize some of the results of [2] to commutative rings R so that every finitely generated projective module is free, called projective free rings. Such rings have no non-trivial idempotents but do not need to be local; indeed, all PIDs are examples.…”

“…In [8,Lemma 2.3] the authors use the resultant of two polynomials to show, over a commutative ring R, for two monic polynomials f, g…”