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Algebra pp 341-394 | Cite as

Bilinear and Quadratic Forms

  • William A. Adkins
  • Steven H. Weintraub
Part of the Graduate Texts in Mathematics book series (GTM, volume 136)

Abstract

Recall that if R is a commutative ring, then Hom R (M, N) denotes the set of all R-module homomorphisms from M to N. It has the structure of an R-module by means of the operations (f+g)(x)=f(x)+g(x) and (af)(x)= a(f(x)) for all x ∈ M, a ∈ R. Moreover, if M=N then Hom R (M, M)= End R (M) is a ring under the multiplication (fg)(x)=f(g(x)). An R-module A, which is also a ring, is called an R-algebra if it satisfies the extra axiom a(xy)=(ax)y=x(ay) for all x,yA and aR. Thus End R(M) is an R-algebra. Recall (Theorem 3.4.11) that if M and N are finitely generated free R-modules (R a commutative ring) of rank m and n respectively, then HomR(M, N) is a free R-module of rank mn.

Keywords

Quadratic Form Bilinear Form Commutative Ring Hermitian Form Invariant Factor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • William A. Adkins
    • 1
  • Steven H. Weintraub
    • 1
  1. 1.Department of MathematicsLouisiana State UniversityBaton RougeUSA

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