not a Field Always because if we take the ideal I = : Z₂ —————→ Z₂ such that y(x) = { 1 if x is odd (0 if x is even ng homomorphism because Criteria for irreducibility Test Fails for f(x)=x6+ 5x³-i but for p= f(x) is irredu. ; The sum of two non-trivial idempotent elements is r because in the ring if we take + more than two idempotent elements in the ring Z₁Z6 ), (, ), (, ) , (, ) multiplicative inverse for (2x+3) in Z₁[x] because (e and b = proper non-trivial maximal ideals in (Z21, > is a maximal ideal in Z2₁, , ) an idempotent in Zn then x is Always an idempotent i 1+x is an idempotent element but x is not X=
not a Field Always because if we take the ideal I = : Z₂ —————→ Z₂ such that y(x) = { 1 if x is odd (0 if x is even ng homomorphism because Criteria for irreducibility Test Fails for f(x)=x6+ 5x³-i but for p= f(x) is irredu. ; The sum of two non-trivial idempotent elements is r because in the ring if we take + more than two idempotent elements in the ring Z₁Z6 ), (, ), (, ) , (, ) multiplicative inverse for (2x+3) in Z₁[x] because (e and b = proper non-trivial maximal ideals in (Z21, > is a maximal ideal in Z2₁, , ) an idempotent in Zn then x is Always an idempotent i 1+x is an idempotent element but x is not X=
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
please solve question 8
![1)
Z₁2/1
Z₁2/1
is a Field.
So if x is even
2) The map y: Z, -----→ Z₂ such that y(x) = {
1 if x is odd
is not a ring homomorphism because
3) Eisenstin Criteria for irreducibility Test Fails for f(x)=x+ 5x³-15x¹+ 15x³+25x² +5x+25
because
but for p=
,f(x) is irreducible using mod p-test.
4) In a ring R; The sum of two non-trivial idempotent elements is not always an
idempotent because in the ring.
idempotent
if we take
is not
"
5) There are more than two idempotent elements in the ring Z6Z6; here are some of
them (, ) , (, ) , ( , ) , (, )
6) There is a multiplicative inverse for (2x+3) in Z₁[x] because (ax+3b) (2x+3)=1_where
A =
and b =
7) There is no proper non-trivial maximal ideals in (Z2₁, , ) is a False statement
because < > is a maximal ideal in Z2₁,
,
)
8) If (1+x) is an idempotent in Z, then x is Always an idempotent is a False statement
because if x=
1+x is an idempotent element but x is not.
3
is not a Field Always because if we take the ideal I =
then](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe8d2d027-204d-466e-850d-579ecb98c79e%2Fac2901fd-d2fd-4edf-96b3-458bb6ad22e2%2Frhpxs5g_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1)
Z₁2/1
Z₁2/1
is a Field.
So if x is even
2) The map y: Z, -----→ Z₂ such that y(x) = {
1 if x is odd
is not a ring homomorphism because
3) Eisenstin Criteria for irreducibility Test Fails for f(x)=x+ 5x³-15x¹+ 15x³+25x² +5x+25
because
but for p=
,f(x) is irreducible using mod p-test.
4) In a ring R; The sum of two non-trivial idempotent elements is not always an
idempotent because in the ring.
idempotent
if we take
is not
"
5) There are more than two idempotent elements in the ring Z6Z6; here are some of
them (, ) , (, ) , ( , ) , (, )
6) There is a multiplicative inverse for (2x+3) in Z₁[x] because (ax+3b) (2x+3)=1_where
A =
and b =
7) There is no proper non-trivial maximal ideals in (Z2₁, , ) is a False statement
because < > is a maximal ideal in Z2₁,
,
)
8) If (1+x) is an idempotent in Z, then x is Always an idempotent is a False statement
because if x=
1+x is an idempotent element but x is not.
3
is not a Field Always because if we take the ideal I =
then
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