On the set of ideals in a ring, divisibility is antisymmetric

Let A and B be ideals in a commutative unital ring R. Show that if A|B and B|A then A = B. Characterize the ideals which divide (1).

If A|B, then B = AC for some C. So B \subseteq A. Likewise, if B|A then A \subseteq B. Thus if A|B and B|A, then A = B.

Suppose A is an ideal such that (1) = AB for some ideal B. Now (1) \subseteq A, so that A = (1). That is, the only unit in the semigroup of ideals in R under ideal multiplication is (1) = R itself.

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