11. Let R and R' be two rings. A mapping f: R→R' iscalled an antihomomorphism, iff(x+y)=f(x) + f(y) and f(xy) = f(y)f(x) x, y € R.Let f, g be two antihomomorphisms of a ring R into R. Prove thatfg: R R is a homomorphism.

Let x, y ∈ R. Then (fg)(x + y) = f(g(x + y)) = f(g(x) + g(y)) = f(g(x)) + f(g(y)) =(fg)(x)+(fg)(y),…

Answered: 11. Let R and R' be two rings. A…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.2: Ring Homomorphisms

Problem 14E: 14. Let be a ring with unity . Verify that the mapping defined by is a homomorphism.

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11. Let R and R' be two rings. A mapping f: R→R' is called an antihomomorphism, if f(x+y)=f(x) + f(y) and f(xy) =f(y)f(x) x, y = R. Let f, g be two antihomomorphisms of a ring R into R. Prove that foRRisa homomorphism