Basic properties of ideal products

Let R be a commutative ring with 1 and let A,B,C \subseteq R be ideals. Prove the following. (1) A(BC) = (AB)C, (2) AB = BA, (3) If a \in A and b \in B then ab \in AB, and (4) AB \subseteq A.


Let a \in A and b \in B. Certainly then by our equivalent characterization of ideal products, ab \in AB.

By this previous exercise, if z \in A(BC) then z = \sum_i a_i (\sum_j b_{i,j}c_{i,j}) = \sum_i \sum_j a_i b_{i,j} c_{i,j}. Each a_ib_{i,j} is in AB, so that (a_ib_{i,j})c_{i,j} \in (AB)C. So A(BC) \subseteq (AB)C. The reverse inclusion is similar.

Suppose \sum a_ib_i \in AB. Since A is an ideal, each a_ib_i is in A. Then \sum a_ib_i \in A.

Since R is commutative, AB = \{\sum_{i=1}^n a_ib_i \ |\ n \in \mathbb{N}, a_i \in A, b_i \in B\} = \{\sum_{i=1}^n b_ia_i \ |\ n \in \mathbb{N}, a_i \in A, b_i \in B\} = BA.

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