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  3. let r be a ring (not necessarily commutative), and

Let R be a ring (not necessarily commutative), and let X denote the set of two-sided...

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Advance Math

Let R be a ring (not necessarily commutative), and let X denotethe set of two-sided ideals

of R.

(i) Show that X is a poset with respect to to set-theoreticinclusion, ⊂.

(ii) Show that with respect to the operations I ∩ J and I + J(candidates for meet and join;

remember that I+J consists of the set of sums, {i + j} where i ∈I and j ∈ J) respectively,

X is a lattice.

(iii) Give an example of a commutative ring for which thecorresponding X is not a Boolean

algebra, and one for which it is.

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