Representativewise multiplication of cosets of an ideal is well-defined

Let R be a ring and let I \subseteq R be an ideal. Show that if a_1 - b_1 \in I and a_2 - b_2 \in I, then a_1a_2 - b_1b_2 \in I.


Since I is an ideal, a_1a_2 - b_1a_2 \in I and b_1a_2 - b_1b_2 \in I. So a_1a_2 - b_1a_2 + b_1a_2 - b_1b_2 = a_1a_2 - b_1b_2 \in I as desired.

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