please solve parts 7, 8 and 9 Question 4: Mark each of the following by True (T) or False (F)1) In a commutative ring with unity every unit is a non-zero-divisor.2) If an ideal I in a commutative ring with unity R contains a unit x then I =R3) In an Integral domain the left cancellation law holds.5) Every finite integral Domain is a field.6) The sum of two idempotent elements is idempotent.7) [123]is a zero divisor in M₂(Z)8) There are 2 maximal ideals in Z₁2 and one maximal ideals in Z89) The polynomial f(x)=x+ 5x³-15x¹+ 15x³+25x² +5x+25 satisfies Eisenstin Criteria forirreducibility Test and therefore it is irreducible over Q.10) If (1+x) is an idempotent in Zn; then (n-x) is an idempotent11) All non-zero elements in Z[i] are non-zero divisors in Z[i]12) In a commutative finite ring R with unity every prime ideal is a maximal ideal

Answered: Image /qna-images/answer/08652d67-7d33-4bdc-bfc9-00e431305900.jpg

Question 4: Mark each of the following by True (T) or False (F) 1) In a commutative ring with unity every unit is a non-zero-divisor 2) If an ideal I in a commutative ring with unity R contains a unit x then I =R . 3) In an Integral domain the left cancellation law holds. 5) Every finite integral Domain is a field. 6) The sum of two idempotent elements is idempotent. 3 7) is a zero divisor in M₂(Z) 26 8) There are 2 maximal ideals in Z₁2 and one maximal ideals in Z8 9) The polynomial f(x)=x+ 5x5-15x¹+ 15x³+25x² +5x+25 satisfies Eisenstin Criteria for irreducibility Test and therefore it is irreducible over Q. 10) If (1+x) is an idempotent in Zn; then (n-x) is an idempotent 11) All non-zero elements in Z[i] are non-zero divisors in Z[i] 12) In a commutative finite ring R with unity every prime ideal is a maximal ideal

Question 4: Mark each of the following by True (T) or False (F) 1) In a commutative ring with unity every unit is a non-zero-divisor 2) If an ideal I in a commutative ring with unity R contains a unit x then I =R . 3) In an Integral domain the left cancellation law holds. 5) Every finite integral Domain is a field. 6) The sum of two idempotent elements is idempotent. 3 7) is a zero divisor in M₂(Z) 26 8) There are 2 maximal ideals in Z₁2 and one maximal ideals in Z8 9) The polynomial f(x)=x+ 5x5-15x¹+ 15x³+25x² +5x+25 satisfies Eisenstin Criteria for irreducibility Test and therefore it is irreducible over Q. 10) If (1+x) is an idempotent in Zn; then (n-x) is an idempotent 11) All non-zero elements in Z[i] are non-zero divisors in Z[i] 12) In a commutative finite ring R with unity every prime ideal is a maximal ideal

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.4: Maximal Ideals (optional)

Problem 26E: . a. Let, and . Show that and are only ideals of and hence is a maximal ideal. b. Show...

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