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
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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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 =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.
7) [123]
is a zero divisor in M₂(Z)
8) There are 2 maximal ideals in Z₁2 and one maximal ideals in Z8
9) The polynomial f(x)=x+ 5x³-15x¹+ 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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0c71d116-0969-4d5d-99cd-20be32a349e1%2F08652d67-7d33-4bdc-bfc9-00e431305900%2F0mjqx2_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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.
7) [123]
is a zero divisor in M₂(Z)
8) There are 2 maximal ideals in Z₁2 and one maximal ideals in Z8
9) The polynomial f(x)=x+ 5x³-15x¹+ 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
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