Bilinear and Quadratic Forms

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,y ∈ A and a ∈ R. 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 Hom_R(M, N) is a free R-module of rank mn.