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wu :: forums    riddles    putnam exam (pure math) (Moderators: Icarus, Grimbal, william wu, towr, Eigenray, SMQ)    Only the trivial ring adds this way « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: Only the trivial ring adds this way  (Read 551 times) ecoist Senior Riddler     Gender: Posts: 405 Only the trivial ring adds this way   « on: Dec 13th, 2006, 2:28pm » Quote Modify Let R be a ring whose additive structure is the rationals under addition modulo 1.  Show that all products in R equal 0. IP Logged Eigenray wu::riddles Moderator Uberpuzzler     Gender: Posts: 1948 Re: Only the trivial ring adds this way   « Reply #1 on: Dec 14th, 2006, 2:26pm » Quote Modify (a/b)*(c/d)  = (a/b)*(bc/bd)  = (a/b)*(c/bd + ... + c/bd)   [b times]  = (a/b)*(c/bd) + ... + (a/b)*(c/bd)  = (a/b + ... + a/b)*(c/bd)  = (a)*(c/bd)  = 0*(c/bd) [ = (0+0)*(c/bd)  = 0*(c/bd) + 0*(c/bd)  = (a/b)*(c/d) + (a/b)*(c/d)]   so (a/b)*(c/d) = 0.   Or: A multiplication on a ring with additive group G is an element of HomZ( G (x)Z G, G).  But if G is divisible and torsion, then G (x)Z G = 0. IP Logged Pages: 1  Reply Notify of replies Send Topic Print Forum Jump: ----------------------------- riddles -----------------------------  - easy   - medium   - hard   - what am i   - what happened   - microsoft   - cs => putnam exam (pure math)   - suggestions, help, and FAQ   - general problem-solving / chatting / whatever ----------------------------- general -----------------------------  - guestbook   - truth   - complex analysis   - wanted   - psychology   - chinese « Previous topic | Next topic »

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