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

Factorization by elementary matrices, null-homotopy and products of exponentials for invertible... Let R be a commutative unital ring. A well-known factorization problem is whether any matrix in $$\mathrm {SL}_n(R)$$ SL 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 $$\mathrm {GL}_n(R)$$ GL n ( R ) as a product of exponentials. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

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

, Volume 9 (3) – Feb 25, 2019
14 pages

/lp/springer-journals/factorization-by-elementary-matrices-null-homotopy-and-products-of-tfmT2yiUQg
Publisher
Springer Journals
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-019-00289-8
Publisher site
See Article on Publisher Site

### Abstract

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

### Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Feb 25, 2019