5. (a) Give the definition of a group.(b) Show that (Z, +) is an abelian group.(c) Let n e N. Then (Matnxn(R), +,') is a ring where + is matrix addition and · is matrixmultiplication.(i) Is there a unity in this ring? (An identity for multiplication.) If yes, give it, if noexplain why not.(ii) Does each non-zero element have an inverse for multiplication? Explain your answercarefully.(d) Give an example of a field with finitely many elements.(It is not necessary to prove your example is a field.)(e*) Let (R, +, ·) be a commutative ring with unity lr # 0R. An element x E R is callednilpotent if there is an integer n EN such thatx" = r.x.. x= 0R.n timesLet S be the subset of R consisting of nilpotent elements.(i) Prove that (S,+, ·) is a subring of (R, +, ').(Hint: Subring criterion and the binomial theorem.)(ii) For each a E S, prove that (1r – x) has a multiplicative inverse in R.

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Answered: 5. (a) Give the definition of a group.…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.1: Definition Of A Group

Problem 51E: Prove that the Cartesian product 24 is an abelian group with respect to the binary operation of...

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