Why is any ring homomorphism a linear transformation? Solution if S,R are rings.

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if S,R are rings. f:R->S is called ring homomorphism if 1 f(x+y)=f(x)+F(y) 2. f(xy)=f(x)f(y) for all x,y belongs to R. and the definition of linear transformation is T:V->W such that following hold. 1. T(v1+v2)= T(v1)+T(v2) for v1,v2 in V 2. T(av)=aT(v) for scalar a. now if x is a constant in case of ring homomorphism condition 2. of ring homoprphim becomes 2. f(ay)=f(a)f(y)=af(y) for all y belongs to R and a is a constant. now, compare both mappings/transformations... f and T. which is exactly same. hence any ring homomorphism a linear transformation

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