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Math Help - Ring

  1. #1
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    Ring

    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 ab=0 for every a,b \in R
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    Quote Originally Posted by Chandru1 View Post

    Let R be a non-zero 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 ab=0

    for every a,b \in R
    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 R^2 =(0): so ab=0, for all a,b \in R and therefore any subgroup of (R,+) is a (right) ideal of R. but R has no non-trivial right ideal. thus (R,+) has no non-trivial subgroup and hence it

    has to be of prime order.

    case 2 R^2 \neq (0): we need a couple of observations first:

    (1) let I=\{x \in R: \ xR=(0) \}. clearly I is a (right) ideal of R. hence either I=R or I=(0). if I=R, then R^2=(0), which is a contradiction. thus I=(0). so xR \neq (0), for all 0 \neq x \in R.

    therefore for any 0 \neq x \in R, we have xR=R.

    (2) now suppose 0 \neq x \in R and let J=\{y \in R: \ \ xy=0 \}. clearly J is a right ideal of R. if J=R, then R=xR=(0), which is nonsense. thus J=(0), i.e. xy=0 implies that x=0 or y=0.

    now let 0 \neq a \in R. by (1): aR=R. so there exists r \in R such that ar = a. thus a(ra-a)=0, which gives us ra=a by (2). the claim is that r=1_R: let K=\{x \in R: \ rx=x \}. obviously K

    is a right ideal of R and K \neq (0), because a \in K. thus K=R, i.e. \forall x \in R: \ rx=x. if x=0, then obviously xr=rx=x=0. if x \neq 0, then since (xr - x)x=x(rx)-x^2=x^2-x^2=0, we

    will get xr=x by (2). therefore r=1_R. now if 0 \neq x \in R, then by (1): xR=R and so xy=1, for some y \in R. similarly yz=1, for some z \in R and thus z=xyz=x(yz)=x. so xy=yx=1.

    this proves that R is a division ring.
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    thanks

    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
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    Quote Originally Posted by Chandru1 View Post

    Ok can u help me with ur email id so that any doubts i can directly email u in(Algebra)
    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?
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