Let R be a ring such that the only right ideals of R are (0) and R. Prove that R is either a division ring or that R is a ring with a prime number of elements in which for every
one of the most notorious problems in Herstein's book as far as i remember! although it's an easy exercise for me now, it gave me a very hard time when i was an innocent undergrad student!
case 1 : so for all and therefore any subgroup of is a (right) ideal of but has no non-trivial right ideal. thus has no non-trivial subgroup and hence it
has to be of prime order.
case 2 : we need a couple of observations first:
(1) let clearly is a (right) ideal of hence either or if then which is a contradiction. thus so for all
therefore for any we have
(2) now suppose and let clearly is a right ideal of if then which is nonsense. thus i.e. implies that or
now let by (1): so there exists such that thus which gives us by (2). the claim is that : let obviously
is a right ideal of and because thus i.e. if then obviously if then since we
will get by (2). therefore now if then by (1): and so for some similarly for some and thus so
this proves that is a division ring.
Hi,
thanks man! herstein problems are the best! Ok can u help me with ur email id so that any doubts i can directly email u in(Algebra)...My email is chandru.mcc@gmail.com and i wud like to have ur help
i'm afraid that would be impossible! i also don't recommand private messeging me because i have a bad habit of replying those messeges quite late and sometimes never!
the best thing to do is to ask your doubts and questions on the forum to increase the chance of getting help and also give other members an opportunity to help or learn.
that's the main idea behind math forums like MHF, isn't it?