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

Hull sinO Bilin 

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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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