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Math Help - Abstract Algebra - Rings and Subrings

  1. #1
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    Abstract Algebra - Rings and Subrings

    Let R be a finite commutative ring, that is if a, b are elements of R, then ab = ba.
    Show that for any t \in R, T = tR = {tr|r \in R} is a subring of R

    Further, if T \neq R, there are r1 \neq r2 such that tr1 = tr2. (Hint: if tr \neq ts for all r, s \in R,
    then #tR = #R. )
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  2. #2
    Senior Member jakncoke's Avatar
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    Re: Abstract Algebra - Rings and Subrings

    Closed under multiplication?  tr_1 * tr_2 = t*(t*r_1*r_2) ,  t*r_1*r_2 \in R so closed under multi.
    Closed under subtraction?  tr_1 - tr_2 = t*(r_1 - r_2) ,  r_1 - r_2 \in R so closed under subtraction
    0*t = 0 \in T

    The third thing. If T = R, then they would have to be the same "size" no? meaning if r_1t = r_2t and  r_1 \not= r_2 then what were originally 2 elements in R collapsed into one element in T (thus making it smaller and thus not equal to R).
    Thanks from jll90
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