# If R is commutative ring with unit element, M be an ideal of R and R/M is finite integral domain then

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If R is commutative ring with unit element, M be an ideal of R and R/M is finite integral domain then
1. M is a maximal ideal of R
2. M is minimal ideal of R
3. M is a vector space
4. M is a coset of R

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Correct Answer - Option 1 : M is a maximal ideal of R

Concept:

If R is a commutative ring then,

ab = ba ∀ a,b ∈ R.

M, which is an ideal of R, will be called the maximal ideal of R,

1) if M ⊂ R, M ≠ R (there is at least one element in R that does not belong to M)

2) There should be no ideal 'N', such that M ⊂ N ⊂ R. (there is no ideal between M and R).

Analysis:

R/M is a field [ every finite integral domain is a field]

R/M is a ring with unity

1 + M ≠ M

i.e., 1 M

Now, one belongs to R, but it does not belong to R.

M ≠ R.

Let I be an ideal of R

Such that M ⊆  I ⊆  R

Let, M ≠ I

a I, such that a M

a + M M

Now, R/M is a field.

Every, non-zero of R/M is revertible

a + M is invertible

b + M R/M such that

(a + M) (b + M) = 1 + M

ab + M = 1 + M

ab – 1 ∈ M ⊆  I      ---(1)

a I, b R

ab I          ---(2)  ( I is an ideal)

From (1) and (2), we can write

ab – (ab – 1) I

1 I

Now, as unity belongs to ideal, so ideal becomes ring

I = R

M is a maximal ideal of R

If R is a commutative ring with unity then every maximal ideal is a prime ideal.