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Paper IPM / M / 9542  


Abstract:  
The ring Z_{2} ×Z_{2}, having only one unit, cannot
be generated by its units. It turns out, in the general theory of rings,
that this is essentially the only example. In this note, we give an
elementary proof of "A finite commutative ring with nonzero identity is
generated by its units if and only if it cannot have Z_{2} ×Z_{2} as a quotient." The proof uses graph theory, and offers, as a
byproduct, that in this case, every element is the sum of at most three
units.
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