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Math Help - [SOLVED] A,B,C right ideals. (A+B)C = AC+BC?

  1. #1
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    [SOLVED] A,B,C right ideals. (A+B)C = AC+BC?

    Hi:
    In a textbook I find: if A, B, C are right ideals in a ring R, then A(B+C) = AB+AC. This I have prove. But I could also prove (A+B)C = AC+BC (1)
    where again A, B, C are right ideals in R. Can (1) be true? Any hint will be welcome.
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  2. #2
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    Quote Originally Posted by ENRIQUESTEFANINI View Post
    Hi:
    In a textbook I find: if A, B, C are right ideals in a ring R, then A(B+C) = AB+AC. This I have prove. But I could also prove (A+B)C = AC+BC (1)
    where again A, B, C are right ideals in R. Can (1) be true? Any hint will be welcome.
    yes it is true even for left ideals. since the product of (left, right or two-sided) ideals I,J is defined to be IJ= \{\sum_{i=1}^n x_iy_i: \ x_i \in I, y_i \in J, \ n \in \mathbb{N} \}, we only need to prove that
    (a+b)c_1 \in AC+BC and ac_1 + b c_2 \in (A+B)C, for all a \in A, \ b \in B, \ c_1,c_2 \in C. these are very easy to prove:

    (a+b)c_1=ac_1 + bc_1 \in AC + BC and ac_1 + bc_2 =(a+0)c_1 + (0+b)c_2 \in (A+B)C.
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  3. #3
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    Ideal products.

    Thanks a lot, NonCommAlg.
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