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Products of involutory matrices over rings. (English) Zbl 0836.15006
An involution in a group $G$ is an element $a$ of order 2. Let $a \in GL_n A$ be a matrix over a commutative ring $A$ with 1 such that $(\text{det} a)^2 = 1$. If $a$ is cyclic, it can be written as a product of at most three involutions. The paper proves that when $A$ satisfies the first Bass stable range condition, then $a$ can be written as a product of at most five involutions. If in addition either $n \le 3$ or $n = 4$ and $\text{det} a = - 1$, then $a$ can be written as a product of at most four involutions. When $A$ is a Dedekind ring of arithmetic type, the number of involutions needed to express $a$ is uniformly bounded for any $n \ge 3$. When $A = \bbfC[x]$ is the polynomial ring with complex coefficients, the number of involutions is unbounded for any $n\ge 2$.
MSC:
15B33Matrices over special rings (quaternions, finite fields, etc.)
15A23Factorization of matrices
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References:
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