For any (unital) exchange ring R whose nitely generated projective modules satisfy the separative cancellation property (A A = A B = B B =) A = B), it is shown that all invertible square matrices over R can be diagonalized by elementary row and column operations. Consequently, the natural homomorphism GL 1 (R) ! K 1 (R) is surjective. In combination with a result of Huaxin Lin, it follows that ...

Let be a semigroup of words over an alphabet . Let consist of those elements of for which every prefix and suffix of belongs to . We show that is a free semigroup. Moreover, is called separative if also the complement is a semigroup. There are uncountably many separative semigroups over , if has at least two letters. We prove that if is separative, then every word has a unique minimum factoriza...

For any (unital) exchange ring R whose finitely generated projective modules satisfy the separative cancellation property (A ⊕ A ∼= A ⊕ B ∼= B ⊕ B =⇒ A ∼= B), it is shown that all invertible square matrices over R can be diagonalized by elementary row and column operations. Consequently, the natural homomorphism GL1(R) → K1(R) is surjective. In combination with a result of Huaxin Lin, it follow...

For any (unital) exchange ring R whose finitely generated projective modules satisfy the separative cancellation property (A ⊕ A ∼= A ⊕ B ∼= B ⊕ B =⇒ A ∼= B), it is shown that all invertible square matrices over R can be diagonalized by elementary row and column operations. Consequently, the natural homomorphism GL1(R) → K1(R) is surjective. In combination with a result of Huaxin Lin, it follow...

Given a unital ring R and a two-sided ideal I of R, we consider the question of determining when a unit of R/I can be lifted to a unit of R. For the wide class of separative exchange ideals I, we show that the only obstruction to lifting invertibles relies on a K-theoretic condition on I. This allows to extend previously known index theories to this context. Using this we can draw consequences ...

If ≤ is a preorder on a set P and p0 ≤ p1, we say that p1 is an extension of p0. Recall that a preorder is separative if and only if whenever p1 is not an extension of p0 there is an extension of p1 which is incompatible with p0. We say that P = (P,≤) is a forcing notion (also forcing poset) if ≤ is a separative preorder with minimal element 0P. Note that if P is separative and p1 p̌0 ∈ Ġ then p...

The replacement of petrochemicals with biobased chemicals requires efficient bioprocesses, biocatalysis, and product recovery. Biocatalysis (e.g., enzyme conversion and fermentation) offers an attractive alternative to chemical processing because biocatalysis utilize renewable feedstocks under benign reaction conditions. One class of chemical products that could be produced in large volumes by ...