Unitary Elements in Simple Artinian Rings

✍ Scribed by C.L. Chuang; P.H. Lee

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 475 KB
• Volume: 176
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1995.1253

✦ Synopsis

The problem of determining when a unitary element is a product of Cayley unitary elements is completely solved for simple artinian rings of characteristic not 2. Theorem 1. Let (D) be a division ring of characteristic not 2 . Suppose that (R=D_{n}) assumes an involution which induces a non-identity involution on (D). Then any unitary element in (R) is a product of two Cayley unitary elements. Theorem 2. Let (F) be a field of characteristic not 2 . Suppose that (R=F_{n}) assumes an involution * of the first kind. Then any unitary element in (R) which is a product of Cayley unitary elements must have determinant 1 . Conversely, any unitary element in (R) of determinant 1 is a product of two Cayley unitary elements, except when (F=G H(3), n=2), and () is given by (\left(\underset{\gamma}{\alpha} \beta{ }_{\delta}^{\beta}\right)^{}=\binom{\alpha-\gamma}{-\beta}). 61945 Academic Press, Inc.