# Ring Theory and Cartesian Product

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If R and S are rings, the cartesian product RxS is a ring too with operations

(r1,s1) + (r2,s2) = (r1+r2,s1+s2)

(r1,s1)*(r2,s2) = (r1*r2,s1*s2)

identity elements

0RxS = (0R,0s) 1RxS = (1R,1S)

and additive inverse -(r,s) = (-r,-s)

If R and S are nontrivial rings, show that RxS has at least 4 idempotent elements

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Ring theory is clarified in this solution.

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