Let R Z6Z What are the ideals of R For each ideal I of R de

Let R = Z/6Z. What are the ideals of R? For each ideal I of R, determine R/I.

Let R be a ring. An additive subgroup I of R that is also strongly closed under multiplication is called an ideal of R.

In the ring Z/6Z, the element r = 3 + 6Z of the ring R = Z/6Z generates a maximal ideal but is not irreducible since

r = r² . The issue is that r is a zero-divisor

Now by homomorphism theorem the ideal in Z/6Z is of the form mz/nz where m is a divisor of n

So the only proper ideals are 2Z/6Z={0,2,4}

and 3Z/6Z = {0,3}

(b) Now R/I = Z/6Z /2 = 1/3

R/I = Z/6Z /4 = 2/3