Factorization by elementary matrices, null-homotopy and products of exponentials for invertible matrices over rings

Let $R$ be a commutative unital ring. A well-known factorization problem is whether any matrix in ${\mathrm{S}\mathrm{L}}_n(R)$ is a product of elementary matrices with entries in $R$. To solve the problem, we use two approaches based on the notion of the Bass stable rank and on construction of a null-homotopy. Special attention is given to the case, where $R$ is a ring or Banach algebra of holomorphic functions. Also, we consider a related problem on representation of a matrix in ${\mathrm{G}\mathrm{L}}_n(R)$ as a product of exponentials.