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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 matrix multiplication. (i) Is there a unity in this ring? (An identity for multiplication.) If yes, give it, if no explain why not. (ii) Does each non-zero element have an inverse for multiplication? Explain your answer carefully. (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 called nilpotent if there is an integer n EN such that x" = r.x.. x= 0R. n times Let 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.

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 matrix multiplication. (i) Is there a unity in this ring? (An identity for multiplication.) If yes, give it, if no explain why not. (ii) Does each non-zero element have an inverse for multiplication? Explain your answer carefully. (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 called nilpotent if there is an integer n EN such that x" = r.x.. x= 0R. n times Let 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.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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 matrix multiplication. (i) Is there a unity in this ring? (An identity for multiplication.) If yes, give it, if no explain why not. (ii) Does each non-zero element have an inverse for multiplication? Explain your answer carefully. (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 called nilpotent if there is an integer n EN such that x" = r.x.. x= 0R. n times Let 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.
Transcribed Image Text: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 matrix multiplication. (i) Is there a unity in this ring? (An identity for multiplication.) If yes, give it, if no explain why not. (ii) Does each non-zero element have an inverse for multiplication? Explain your answer carefully. (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 called nilpotent if there is an integer n EN such that x" = r.x.. x= 0R. n times Let 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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