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Definition: Let R and R' be rings. The product set RxR' is a ring called the product ring, with component-wise addition and multiplication: (x,x')+(y,y')=(x+y,x'+y') and (x,x')(y,y')=(xy,x'y'). The additive and multiplicative identities in RxR' are (0,0) and (1,1), respectively.
Source: Algebra, second edition by Michael Artin