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Let R be a commutative unital ring. A well-known factorization problem is whether any matrix in $$\mathrm {SL}_n(R)$$SLn(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 $$\mathrm {GL}_n(R)$$GLn(R) as a product of exponentials.

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