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Nontrivial Finite Rings with No Zero Divisors
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[QUOTE="fresh_42, post: 5855893, member: 572553"] Why do left and right identities have to be the same? I could follow you until - given an element ##r \neq 0## - there are elements ##x_r## and ##y_r## with ##rx_r =r = y_rr##. The index is necessary as up to this point, those elements still depend on the given ##r##. Also isn't clear whether we may assume the existence of a unity element ##1## or not. E.g., how about the ring of all polynomials over ##\mathbb{Z}_3## with zero absolute term and ##x^2=x##, i.e. ##x\,\mid \,p(x)\,##? And I haven't understood the part, where you show that ##x_r=x_s##. This leaves me with three questions: [LIST=1] [*]##1 \in R\,##? [*]##x_r = y_r\,##? [*]##x_r = x_s## and with it ##y_r=y_s\,##? [/LIST] So isn't clear to me and the third part is a bit too hand waving. I think at some place associativity is needed (which also isn't mentioned as a given property). Without indexing the right identity ##x_r## and the left identity ##y_r## it is a bit too foggy for my taste. [/QUOTE]
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Nontrivial Finite Rings with No Zero Divisors
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