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 ${SL}_n\left(R\right)$ 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 ${GL}_n\left(R\right)$ as a product of exponentials.