Titlebook: Algebra; Some Recent Advances I. B. S. Passi Book 1999 Hindustan Book Agency (India) and Indian National Science Academy 1999 Area.Volume.a

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Alternative Loop Rings and Related Topics,, see Definition 3.1). The . of . over . was introduced in 1944 by R.H. Bruck (1944) as a means to obtain a family of examples of nonassociative algebras and is defined in a way similar to that of a group algebra; i.e., as the free A-module with basis ., with a multiplication induced distributively from the operation in .

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Md Musfique Anwar,Jianxin Li,Chengfei Liu(1981) gives some later developments (see also the books of Sehgal, 1989 and Karpilovsky, 1989). In this article our main aim is to survey the more recent developments. In § 1 we review the case when . is a field and in §2 the case of the integral group ring is considered.

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Xiu Susie Fang,Xianzhi Wang,Quan Z. Shenger fields, a main step in the proof of these conjectures is a classification theorem of hermitian forms over involutorial division algebras defined over fields of virtual cohomological dimension ≤ 2, which is described in § 6 and § 7.

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Lei Li,Xiaofang Zhou,Kevin Zhengtally, to the construction in ([PI]) of non diagonalisable, (in fact indecomposable), non singular symmetric 4 × 4 matrices of determinant one over the polynomial ring in two variables over the field of real numbers, producing remarkable counter examples to the so called quadratic analogue of Serre’

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Unit Groups of Group Rings,(1981) gives some later developments (see also the books of Sehgal, 1989 and Karpilovsky, 1989). In this article our main aim is to survey the more recent developments. In § 1 we review the case when . is a field and in §2 the case of the integral group ring is considered.

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